the aim of this paper is to provide a short summary of what we know about the group field theory ( gft ) formalism for quantum gravity and its longer term goals . for greater agility of presentation and reading ,the format is that of a faq : we outline tentative and provisional answers to some basic questions about the gft formalism , which aim at reflecting the current understanding of the subject , from the strict point of view of the authors . having clarified our aims , let us clarify what this paper is _ not _ meant to be .first , it is not meant to be a technical paper dwelling into any of the details of the formalism , or explaining in detail any of the recent results . for thiswe will refer to the relevant literature .second , it is not a pedagogical , extensive introduction to gfts .for early introductions , the reader should go to , and for the general idea of tensor models one should look at the early papers and at the literature on matrix models . for a more extensive , and almost up to date introduction to gfts for quantum gravity, we refer to .third , it is not a proper review of the subject nor of its recent developments , which are many and important . for many of the recent results on ( colored )tensor models , a complete and up - to - date ( and beautifully written ) review is . for many of the recent results that relate to the geometry of group field theories , we refer to the review . for a discussion of spin foam models , in particular the recent ones , and thus some important aspects of the corresponding group field theories , we refer to the many reviews on spin foams .many other results , concerning for example gft and tensor model renormalization can unfortunately only be found in the original papers .\(i ) * what is a group field theory ? *  group field theories are a specific class of tensor quantum field theories which generalize matrix models to higher dimensions .the fundamental variables are in fact higher rank _ tensors _ formally denoted by where the indices label points in the direct product of d domain spaces .the domain space could be a manifold or a finite set .the dynamics of tensor models is governed by a classical action characterized by a peculiar pairing of the indices in the interaction term : where is the set of indices of the copy of the tensor in the monomial of degree , with .for example the action for the independent identically distributed 3-tensor model is [ tensor ] s(t ) = 12 _ i , j , k t_ijk t_kji - _ ijklmn t_ijkt_klm t_mjn t_nli a tensor can be graphically represented as a d - valent node with an index on each line , or dually as a ( d-1)-simplex with an index on each ( d-2)-face .the interaction term pairs the indices along the links of a network with d+1 d - valent nodes ; dually it patterns the glueing of d+1 ( d-1)-simplices along common faces to form a d - simplex .the kinetric term describes instead the identification of two nodes or ( d-1)-simplices .different choices of kinetic and interaction terms , are possible and define different models in the same class .the peculiar interaction combinatorics of any tensor model , shows up in the perturbative expansion of the theory in feynman diagrams }\,=\,\sum_{\gamma}\,\frac{\lambda^v}{sym[\gamma]}\,z(\gamma),\ ] ] where is the number of interaction vertices in the feynman graph , $ ] is a symmetry factor for the graph and the corresponding feynman amplitude . by constructionthe feynman diagrams will be represented as stranded diagrams dual to cellular ( simplicial ) complexes of arbitrary topology , obtained by arbitrary glueings of d - simplices along ( d-1)-faces .tensors models are thus a straightforward combinatorial generalization of matrix models for 2d quantum gravity . in matrix models , the feynman amplitudes associated to each diagram can be re - interpreted in terms of a discretization of 2d general relativity on an equilateral triangulation , and the overall sum as a generalization of a lattice path integral for 2d gravity to arbitrary triangulations and arbitrary topologies . among the key results in matrix models are the large n topological expansion dominated by trivial topologies ( where n is the dimension of the matrix ) and the identification of critical behaviour for some value of the coupling constant .this critical behaviour defines the continuum limit , shown by various methods to match the effective continuum dynamics to 2d liouville quantum gravity .tensors models try to export these ideas and results to higher dimensions .however , while even simple tensor models of the type ( [ tensor ] ) have an interest for quantum gravity , as they relate directly to the approach of dynamical triangulations , both the richness of spacetime geometry in higher dimensions ( especially in 4 dimensions ) and the results obtained by other approaches to quantum gravity suggest that a richer set of data and symmetries should be added to the tensors , and that a more involved dynamics should be chosen , to have a better chance to describe quantum geometry and gravity .group field theories ( gft ) is the name used , in fact , for those tensor models where the domain space is chosen to be the local gauge group of gravity ( i.e. the lorentz group or its euclidean counterpart ) , so that tensors thus turn into fields . an additional symmetry is invoked which captures local gauge invariance : and , possibly , kinetic and interaction terms are characterized by non - trivial kernels . in turn , alternative representations in terms of group representations or as non - commutative quantum field theories over , where is the dimension of , become available , thanks to the group structure and tools from representation theory and non - commutative geometry .an example is the boulatov model for 3d riemanian quantum gravity , where and with action : s_3d[]= 12 ^3 _ 123_321 - ^6 _ 123_345_526_641 [ boulatov ] where is a short notation for and is the haar measure on the gauge group .the gft formalism thus attempts to _ define _ a theory of quantum spacetime as a superposition of discrete spaces , each generated as a possible interaction process of fundamental building blocks , tentative quanta of space , by incorporating into the tensor model framework lessons and insights of other approaches to quantum geometry quantum regge calculus , quantum simplicial geometry and loop quantum gravity ( lqg ) , as well as mathematical tools taken from non - commutative geometry :  * the insight from quantum simplicial geometry suggests the variables needed to describe a ` quantum geometric ( d-1)-simplex ' which has to become the fundamental atom of quantum space , the basic ` quantum ' in our enriched tensor model .* a characterization of the geometry and of the whole kinematical phase space to be associated to a simplex is also provided by the analysis of discretized gravity actions in various dimensions and in various formulations . in particular , 1st order formulations of gravity in terms of connection and ( conjugate ) vielbein variables suggest to use group and lie algebra manifolds to define the phase space of such simplices , and the domain spaces of tensor models , leading then to group field theories .* loop quantum gravity provides further justification for the kinematical structures used for defining the hilbert space of quantum states of these models , several insights on their interpretation , and a main source of inspiration for constructing models and specifying the correct dynamics . *simplicial gravity path integrals , and in particular quantum regge calculus , provide a template to analyze the corresponding feynman amplitudes , and to evaluate to what extent they capture correctly the dynamics of discrete geometry that can be associated to each feynman diagram / simplicial complex .* tools from non - commutative geometry then provide the means to define different representations of the same models , and to analyze their properties .\(ii ) * how does the gft formalism relate to loop quantum gravity ? *  polynomial gauge invariant observables of the group field theory correspond to functions of a finite number of gauge group elements associated to the links of a ( d - valent ) graph ( plus possible additional variables ) satisfying a gauge invariance condition at each node .a basis of such observables is written in terms of so - called spin networks functionals , generically labelled by a graph with edges and vertices respectively group representations and invariant tensors .gft n - point functions can thus be expressed in terms of lqg - like states .the exact matching holds for some models , at least in representation space ( and as far as the spectrum of some quantum geometric operators is concerned ) .intuitively , gfts provide a ` second quantized ' framework for lqg : spin network vertices become the quanta of a gft field , and the lqg wave function associated to an individual vertex is turned to a field operator that creates / annihilates spin network vertices from / to a ` no - space ' vacuum , where no geometrical nor topological structure is present .it is difficult , however , to go beyond this intuitive picture and construct a proper fock space of spin network vertices : to do so , one has to face the issue of choosing a statistics for them , which is both a very difficult and very exotic question that has never been considered in the lqg literature .as far as the quantum dynamics of lqg is concerned , the main link lies in the structure of the gft feynman amplitudes . these can indeed be written in terms of _ spin foam models _ , which show up in lqg as a natural arena to encode the hamiltonian evolution of spin - networks as a sum over histories .a ` spin foam ' is a higher dimensional analogue of a spin - network ( a ` history ' of spin - network evolution ) : it is a system of branching surfaces taking the form of a 2-complex , with polygonal faces labeled by group representations and edges labeled by intertwiners . to each gft feynman diagram corresponds a 2-complex : vertices and edges of corresponds to vertices and edges of and the loops of strands on corresponds to the 2d faces of .one can show that the gft amplitudes ( in a specific representation ) takes the generic form of ` spin foam amplitudes ' : i_g = _ j_f , _ e _ f a_f(j_f ) _ e a_e(j_f , _ e ) _ v a_v(j_f , _ e ) characterized by a choice of local amplitudes and assigned to the faces , edges and vertices of the 2-complex defined by .for example , the feynman amplitudes of the boulatov model ( [ boulatov ] ) give the ponzano - regge model for 3d quantum gravity . this duality between gft and spin foam amplitudesis general : the amplitudes of any spin foam model can be obtained as the feynman amplitude of a gft .  in a covariant ,sum - over - histories formulation of loop quantum gravity , both the combinatorial and algebraic data that specify a quantum state of geometry are randomized with a quantum probability amplitude in order to define the full quantum theory .gfts thus provide a definition of this quantum dynamics , in the form of a quantum field theory expansion in possible interaction processes for the same building blocks of quantum geometry , which includes a sum over all topologies .the main object of interest for the lqg dynamics is the physical inner product between states , projecting onto the kernel of the hamiltonian constraint .indications of where this object should be found in the gft formalism stem from the interpretation of gft as a second quantized formalism for spin network vertices .first , one would expect that some form of hamiltonian constraint acting on spin network vertices can already be identified at the level of the gft classical action and equations of motion .second , as we mentioned , the dynamics encoded into a single evolution history of a spin network state can be found in all its details in the form of a gft feynman amplitude .obviously , from the field theory perspective , this is but a tiny corner of the true quantum dynamics , which has to be looked for in the schwinger - dyson equations ( sde ) for the n - point functions of the gft model . in other words , one would expect that the sde for polynomial field correlations , that is spin network functionals , would admit ( at least in a regime suppressing topology change ) an interpretation and a transcription as hamiltonian and diffeomorphism constraints of a quantum gravity theory , and thus provide the definition of the physical inner product .this is indeed what happens , in the appropriate limit , in the simple case of matrix models . in particular, this should be true in the continuum limit of the gft model , which is necessarily a non - perturbative domain .the same continuum dynamics can be also looked for at the level of effective actions in this continuum limit , at the level of symmetries satisfied by the model and , if a phase transition is part of the picture , in the nature of the critical exponents for the scaling of interesting observables .\(iii ) * why is the gft formalism useful to lqg ? *this way of defining the quantum dynamics is advantageous , from a purely lqg perspective , for a variety of reasons .the main one is that a field theory framework is the most convenient one for dealing with the infinite number of degrees of freedom which should ultimately be present in the full continuum theory : a generic continuum geometry will in fact be captured by a highly complicated ( superposition of ) spin network state(s ) , that is , in gft language , a hugely populated many - particle state .furthermore , contrary to direct attempts at the definition of the continuum gravitational dynamics from the canonical quantum gravity perspective , the gft formalism offers a compact , but in principle complete definition of the theory : it is given by the path integral of the gft model .the difficulty is of course to go beyond the formal definition , prove its well - posedness , and extract interesting physics from it .the gft definition also opens the door to the application of more or less standard quantum and statistical field theory tools ( and ideas ) to the study of the same dynamics , a very powerful machinery indeed .last , it permits to lqg structures and results to force their way out of the canonical quantization of continuum gravity cage , which is not only technically extremely challenging , but also suspicious for a variety of reasons .several arguments indeed ( very different in nature , each not conclusive , and still each very reasonable ) can be put forward suggesting that general relativity ( gr ) is but an effective field theory , and that it should not be quantized as such , but only emerge in some corner of a fundamental theory defined in altogether different terms .gfts allow in principle to test and , if realized , to describe in detail this scenario .\(iv ) * how to define the right quantum dynamics of geometry as a gft ?*  given the ingredients entering the very definition of a gft model , the most natural way of proceeding to its construction relies mainly on quantum simplicial geometry .it amounts to answering three questions :  a.  what is a quantum geometric ( d-1)-simplex ?b.  what is a quantum geometric d - simplex , or , equivalently , how to glue d+1 ( d-1)-simplices to form a d - simplex ? c.how does the geometric information of one d - simplex propagate to a neighboring one ?the answer to a ) dictates the nature and properties of the gft field and its variables .the answer to b ) indicates what interaction kernel one should use to define the classical action of the corresponding gft . the answer to c ) gives the form of the kinetic term of the action , that is , the propagator of the field theory .having all the ingredients at hand , one can write down the gft action , and thus the full model , in perturbative expansion. a first source of inspiration for answering these questions comes from ideas of geometric quantization , also at the roots of spin foam models .let us see this in the case of the boulatov model ( [ boulatov ] ) : upon peter - weyl expansion into irreducible representations labelled by , the gauge invariant field is expressed in terms of three valent - invariant tensors \ ] ] in this picture , the boulatov field represents a quantum triangle with fixed lengths : each represents the hilbert space of states of a quantized vector of length , upon geometric quantization that promotes the classical edge vector into the generators ; the classical closure condition for a triangle then projects into the subspace of invariant tensors .geometric quantization here also gives us the glueing rules for triangles as traces of tensors .in such a representation of group field theory , the feynman amplitudes take the form of spin foam models .the challenge has been to export these ideas to four dimensions for the construction of realistic quantum gravity models .an elegant , complementary approach , which has the advantage of making the geometry content of gft more manifest , has recently been proved fruitful . in this approach ,the field variables are data encoding classical simplicial geometry ( edge vectors in 3d , area bivectors in 4d ... ) , and their quantization results in a non - commutative star - product on the space of fields .this non - commutative structure rests on the phase space structure of the discrete classical theories these gft models are set to quantize , usually given by the cotangent bundle of a group manifold , with configuration space being a lie group and the dual space being the corresponding lie algebra ; it thus encodes the classical non - commutativity of the ` momentum ' space of the classical theory . in the case of the model ( [ boulatov ] ) , upon so - called group fourier transform ( x_1,x_3):= ^3 ( g_1 , g_3 ) ^i x_1 g_1 ^ix_3 g_3 the model is expressed in terms of fields on copies of endowed with a non - commutative star - product ( dual to group convolution ) .gauge invariance translates into a closure condition for the variables , thus naturally interpreted as the edge vectors of a ( non - commutative ) triangle .glueing rules for the triangles are then simply dictated by the star product , in a way that identifies the edge vectors of the common edges by means of non - commutative -delta functions .this result in the following expression for the action ( [ boulatov ] ) : [ action in x ] s [ ] = 12 ^3 _ 123 _ 321 - ^6 _ 123 _ 345_526 _ 641 where the star - product relates pairwise the variables with the same index .in such a representation of group field theory , the feynman amplitudes take the form of simplicial gravity path integrals .remarkably , this approach easily extends to 4d gravity models .\(v ) * what are the symmetries of a gft model and what are their consequences ? *  we start entering a less explored territory , despite the great importance of the issue .symmetries are in fact crucial , in a field theory context , to constrain model building and understanding the true nature of the models , to define appropriate observables , to identify universality classes of critical behaviour , to distinguish different phases of the system , and so on .  up to now , only a few symmetries of gft models for topological and , even less , gravity 4d models are understood and under some control .the gft implementation of the symmetry which becomes , at the level of the gft feynman amplitudes , the local gauge invariance of simplicial path integrals is well - understood .recently , also the gft symmetry corresponding to 3d simplicial diffeos ( after gauge invariance is imposed ) and to translation invariance in bf models has been uncovered .interestingly , all these transformations , which are local symmetries in the simplicial path integral , are global transformations of the gft field , from a field - theoretic point of view .moreover , diffeos / translations are implemented as _ quantum group _ symmetries .these are clearly at least partially broken in 4d gravity models , but the details of this breaking have not been elucidated yet .combinatorial symmetries with respect to permutations of the arguments of the gft field have also been considered , but their implications beyond those on the structure of the feynman diagrams have not been explored .a more careful study of gft symmetries and of their consequences in gft models , the 4d gravity ones in particular , is certainly needed .one class of transformations that would be interesting to identify is the gft and tensor models analogue of the unitary matrix transformations that allow the re - writing of matrix models in terms of matrix eigenvalues ( the true degrees of freedom ) .this re - writing is crucial for many of the key results in matrix models and one could expect a similar role in tensor models and gfts .\(vi ) * can one control the sum over diagrams and topologies ?*  the perturbative sum is in fact pretty wild , including both simplicial manifolds of arbitrary topology and pseudo - manifolds .a number of works have been devoted to a better understanding of the properties of the generated complexes .an important development was the definition of a gft analogue of the large - n expansion of matrix models , dominated by a particular class of triangulations of the sphere , for simple tensor models and topological gfts in any dimension .the gft analogue of the size of the tensors is a cut - off in the momenta ( group representations ) .this expansion is yet to be understood in its physical significance and in its subdominant terms , but it opens the way for several applications , both in quantum gravity and statistical physics .mathematically , it is complemented by a variety of results on the scaling behaviour with the cut - off of the amplitudes of topological bf models ( and some more limited results on 4d gravity models ) , including rigorous power counting results . for some models , it was also possible to prove the borel summability of the whole perturbative series .( vii ) * is the gft perturbative expansion physically useful ? *  from the gft point of view , the discrete data appearing in the gft action and amplitudes are physical , in the sense that they correspond to true degrees of freedom of the theory , and not mere discretization artifacts. however , they are obviously very different from the variables we are used to adopt for describing continuum spacetime physics . from simplicial gravity , we know that these data correspond to ( piecewise - flat ) singular continuum geometries , which in turn can be used to reconstruct smooth geometries in the limit in which large numbers of the same data are considered ( at least at the classical level ) .this points to the non - perturbative sector of the gft formalism as the locus of continuum geometric physics ( as indeed shown explicitly in the simple case of matrix models ) .however , being physical , even the data associated to finite ( sums of ) simplicial complexes ( e.g. those obtained truncating the gft perturbative sum to finite order ) could in principle be used to describe physical phenomena , in the same way in which a finite number of particles could be used , in some limited circumstances , to mimic the behaviour of a fluid . what is hard, however , is to pinpoint what these circumstances are , exactly . in general, being simplicial gravity path integrals , one would say that the feynman amplitudes of gfts describe correctly gravitational phenomena to the extent in which piecewise - flat regge geometries can be used to approximate them .this means we could take advantage of the expertise developed , in the use of such approximations , in numerical relativity .however , one could expect that the perturbative gft expansion is more and more physically relevant , and convenient as a mathematical language , the closer we are , physically , to the spacetime configuration represented by the gft perturbative vacuum , which is unfortunately a very degenerate configuration in which no geometry and no space is actually present .\(viii ) * why does one want to have a renormalizable gft ?is it even possible ? *  being a field theory , and aiming at a fundamental description of space and time down to the planck scale , a gft model should be renormalizable .non - renormalizability would instead implies that the gft model considered can at best be an effective description of a more fundamental theory .the renormalization group flow of a given gft will tell us what type of interaction ( thus , which dynamics ) is relevant at various scales .  the question becomes then whether one can hope to obtain such a renormalizable gft .the main source of doubt probably comes from the non - renormalizability of gravity around minkowski spacetime .however gfts are background independent formulations of gravity ( hopefully ) , not limited to fluctuations of the metric around a given continuum spacetime geometry , and their perturbative expansion is formulated around a very different vacuum , one that corresponds to no - space at all . being formulated in terms of fundamental structures that are either non - geometric in nature or anyway only associated to discrete geometries , it is not obvious what effective continuum degrees of freedom it captures . while this is obviously a difficulty in relating gfts to continuum physics , it also warns us against the direct application of conventional wisdom about continuum perturbative quantum gravity .studies of tensor models and gft renormalization are just beginning , but already one example of a fully renormalizable non - trivial tensor model in 4 dimensions , though much simpler than full - blown gfts for 4d gravity , has been recently exhibited .\(ix ) * where should one look for continuum gravity ? *  the dynamics of smooth geometries should emerge as an effective dynamics for many degrees of freedom , i .. e from large boundary graph - based states and the re - summation of infinite feynman diagrams .in other words , continuum gravitational physics resides in the non - perturbative sector of the current definition of gfts .this means looking for different vacua of the fundamental gft and for the dynamics around them , possibly accompanied by ` large - scale ' and semi - classical approximations .notice , however , that not only the semi - classical limit is a distinct limit from the continuum one ( the latter being naturally a limit of increasing number of degrees of freedom , regardless of their quantum or classical nature ) , but the appropriate continuum limit could even emerge because of the quantum properties of the fundamental degrees of freedom , which should not be washed out beforehand .the example that comes to mind is that of quantum liquids .obviously , this is _ terra incognita _ , at present , so it is hard to do more than speculations and some educated guesses .  if more and more interacting degrees of freedom are to be taken into account on the way to the continuum limit , and if the theory has to be trusted over a wide range of scales ( as a fundamental theory of quantum gravity should be ) , a phase transition should occur .physically , the occurrence of such a phase transition switching from microscopic excitations to collective ones , would correspond to a ` geometrogenesis ' , the true birth of continuum spacetime and geometry from a fundamental non - geometric phase .this scenario has been proposed repeatedly in the gft context and outside it .we are gaining experience of phase transitions following from the resummation of the perturbative series in the large - n limit , in simple and less simple tensor models .but more work is needed to establish such scenario in more involved gft models , and to understand its physical significance .\(x ) * can we do physics with gfts ? *  in light of the above scenario , the answer is unfortunately ` not yet ' , given the current status of our understanding of the subject .the task is to improve our control over the non - perturbative sector of gft and our tools for studying phase transitions in gfts . in the meantime , the exploration of approximation methods for gft has started , to study the effective dynamics for fluctuations around non - trivial gft field configurations by expanding the gft action around them , interpreted as non - trivial quantum spacetime backgrounds .the results ( including the derivation of effective geometrodynamic equations , effective hamiltonian constraint operators , emergent non - commutative matter field theories ) are very interesting , but there is clearly still a long way to go . in parallel ,the construction of simplified gft - like models has also started , in particular models with possible cosmological interpretation and application .the idea is that such simplified models could either be derived from or at least guide the mean field theory treatment of fundamental gft models .again , this is just a beginning .we have answered , to the best of our current understanding of the subject , some questions about the gft approach to quantum gravity , its relation with lqg and other approaches , and its present status . in the process , we have mentioned many of the important results obtained recently in this context , and outlined some direction for future work .we hope to have well represented the fast growth of this area of research , and convinced the reader of its promise . at the same time , we hope it stands clear that much more work is needed to address many of these questions fully , and in particular those whose answer will really establish gfts as a solid candidate formalism for quantum gravity .   99 d. oriti , ed ., _ approaches to quantum gravity _ , cambridge university press , cambridge ( 2009 ) l. freidel , int.j.phys .* 44 * , 1769 - 1783 , ( 2005 ) arxiv : hep - th/0505016 .d. oriti , in , [ arxiv : gr - qc/0607032 ] ; d. oriti , arxiv:0912.2441 [ hep - th ] ; d. oriti , in _ quantum gravity _ , b. fauser , j. tolksdorf , e. zeidler , eds . ,birkhaeuser , basel , ( 2007 ) , [ arxiv : gr - qc/0512103 ] m. gross , nucl .. suppl . *25a * , 144 - 149 , ( 1992 ) ; j. ambjorn , b. durhuus , t. jonsson , mod .. lett . * a6 * , 1133 - 1146 , ( 1991 ) ; n. sasakura , mod.phys.lett . a6 ( 1991 ) 2613 - 2624 f. david , nucl .b257 , * 45 * ( 1985 ) ; p. ginsparg , arxiv : hep - th/9112013 ; p. di francesco , p. ginsparg , j. zinn - justin , phys .( 1995 ) 1 - 133,hep - th/9306153 ; p. ginsparg , g. moore , hep - th/9304011 d. oriti , in g. ellis , j. murugan , a. weltman ( eds ) , _ foundations of space and time _ , cambridge university press , cambridge ( 2011 ) , arxiv:1110.5606 [ hep - th ] r. gurau , j. ryan , sigma , arxiv:1109.4812 [ hep - th ] a. baratin , d. oriti , sigma , to appear a. perez , class .. grav . * 20 * , r43 , ( 2003 ) , [ arxiv : gr - qc/0301113 ] ; d. oriti , gr - qc/0311066 ; j. c. baez , class .grav . 15 , 1827 ( 1998 ) [ arxiv : gr - qc/9709052 ] , d. v boulatov , mod .a*7*:1629 - 1646 ( 1992 ) , arxiv : hep - th/9202074 . c. rovelli , _ quantum gravity _ , cambridge university press , cambridge , ( 2006 ) .m. p. reisenberger and c. rovelli , class .18 , 121 ( 2001 ) [ arxiv : gr - qc/0002095 ] .h. ooguri , prog.theor.phys.89:1-22,1993 , arxiv : hep - th/9210028 . j. w. barrett , l. crane , j. math . phys .39 , 3296 ( 1998 ) , arxiv : gr - qc/9709028 .a. baratin , d. oriti , phys .* 105 * ( 2010 ) 221302 , arxiv:1002.4723 [ hep - th ] .a. baratin , b. dittrich , d. oriti , j. tambornino , class .( 2011 ) 175011 , arxiv:1004.3450 [ hep - th ] , a. baratin , d. oriti , to appear in new j. phys ., arxiv:1108.1178 [ gr - qc ] .a. baratin , d. oriti , arxiv:1111.5842[hep - th ] .l. freidel , e. livine , class .* 23 * , 2021(2006 ) , arxiv : hep - th/0502106 .s. majid , _ foundations of quantum group theory _ , cambridge university press ( 1995 ) .h. ooguri , mod .lett . a * 7 * , 2799 ( 1992 ) , hep - th/9205090 .f. girelli , e. livine , class .27 , 245018 ( 2010 ) , arxiv:1001.2919 .a. baratin , f. girelli , d. oriti , phys .d * 83 * 104051 ( 2011 ) , arxiv:1101.0590 [ hep - th ] .j. ben geloun , arxiv:1107.3122 .r. de pietri and c. petronio , j. math .41 , 6671 ( 2000 ) [ arxiv : gr - qc/0004045 ] .r. gurau , class .27 , 235023 ( 2010 ) , arxiv:1006.0714 [ hep - th ] .r. gurau , annales henri poincare 12:829 - 847 ( 2011 ) , arxiv:1011.2726 [ gr - qc ] .r. gurau and v. rivasseau , europhys.lett.95:50004,201 , arxiv:1101.4182[gr - qc ] l.freidel , r. gurau , d. oriti , phys.rev .d80:044007 ( 2009 ) , arxiv:0905.3772[hep - th ] .j. ben geloun , t. krajewski , j. magnen , v. rivasseau , arxiv:1002.3592[hep - th ] .t. krajewski , j. magnen , v. rivasseau , a. tanasa , p. vitale , phys .d82:124069 ( 2010 ) , arxiv:1007.3150 [ gr - qc ] .j. ben geloun , v. bonzom , int.j.theor.phys.50:2819-2841 ( 2011 ) , arxiv:1101.4294[hep - th ] .l. freidel , d. louapre , phys .d68 ( 2003 ) 104004 , arxiv : hep - th/0211026[hep - th ] .v. rivasseau , pos cncfg2010 ( 2010 ) 004 , arxiv:1103.1900 [ gr - qc ] .j. ben geloun , v. rivasseau arxiv:1111.4997 [ hep - th ] .t. konopka , f. markopoulou , l. smolin , hep - th/0611197 d. oriti , j.phys.conf.ser .67 ( 2007 ) 012052 , hep - th/0612301 d. oriti , proceedings of science , [ arxiv:0710.3276 ] ; l. sindoni , arxiv:1105.5687 [ gr - qc ] ; d. oriti , arxiv:1107.4534 [ physics.pop-ph ] n. seiberg , hep - th/0601234 v. bonzom , r. gurau , a. riello , v. rivasseau , nucl.phys .b853 ( 2011 ) 174 - 195 , arxiv:1105.3122 [ hep - th ] d. benedetti , r. gurau , nucl.phys . b855 ( 2012 ) 420 - 437 , arxiv:1108.5389 [ hep - th ] d. oriti , l. sindoni , new j.phys . 13( 2011 ) 025006 , arxiv:1010.5149 [ gr - qc ] w. fairbairn and e. livine , class .* 24 * , 5277 ( 2007 ) [ arxiv : gr - qc/0702125 ] f. girelli , e. livine and d. oriti , phys .d81 , 024015 ( 2010 ) , [ arxiv : 0903.3475 [ gr - qc ] ] e.livine , d. oriti , j. ryan , to appear in class.quant.grav ., arxiv:1104.5509 [ gr - qc ] g. calcagni , s. gielen , d. oriti , to appear

Abstract: we provide a short and non - technical summary of our current knowledge and some possible perspectives on the group field theory formalism for quantum gravity , in the form of a ( partial ) faq ( with answers ) . some of the questions and answers relate to aspects of the formalism that concern loop quantum gravity . this summary also aims at giving a brief , rough guide to the recent literature on group field theory ( and tensor models ) .


we consider the motion of an incompressible fluid in a bounded two - dimensional domain with some nonlinear boundary conditions , specified as the _ slip boundary condition of friction type _( sbcf ) or the _ leak boundary condition of friction type _ ( lbcf ) .these boundary conditions were introduced by h. fujita in , and subsequently , many studies have focused on the properties of the solution , for example , existence , uniqueness , regularity , and continuous dependence on data , for the stokes and navier - stokes equations under such boundary conditions .details can be referred to in itself or in , , , and , among others .similar types of nonlinear boundary condition , such as subdifferential boundary condition or tresca boundary condition , have been reported in , , and , among others .the frictional boundary conditions under consideration have been successfully applied to some flow phenomena in environmental and medical problems such as oil flow over or beneath sand layers and blood flow in the thoracic aorta .such applications have been discussed in , , , and . in these works ,the finite difference method is used for discretization , and theoretical considerations such as convergence are not addressed .  on the other hand ,few studies have focused on the theoretical analysis of numerical methods for these boundary conditions , even if restricted to the stokes problem .for example , li and li proposed a finite element approximation combined with a penalty method for the stokes equation with sbcf .they proved the optimal order error estimate ; however , they did not focus on a numerical realization of their finite element approximation .the purpose of this work is to construct a comprehensive theory of the finite element method applied to flow problems with sbcf and lbcf , including all of the existence and uniqueness result , error analysis , and numerical implementation . in doing so , herein, we restrict our consideration to the stationary stokes equation in a two - dimensional polygon .the remainder of this paper is organized as follows . in section 2 ,we review the results for the continuous problems described in .weak formulations by an elliptic variational inequality for sbcf and lbcf are also presented . in section 3 , we prepare the finite element framework using the so - called p2/p1 element , and state several technical lemmas .section 4 is devoted to the study of approximate problems for sbcf .we propose the discretized variational inequality problem , proving the existence and uniqueness of a solution . in the error analysis, we first derive a primitive result of the convergence rate under the - regularity assumption with .second , we show that it is improved to under the additional hypothesis of good behavior of the sign of the tangential velocity component on the boundary where sbcf is imposed . a sufficient condition to obtain , which is of optimal order when , is also considered .finally , we propose an iterative uzawa - type algorithm to perform numerical computations , and prove that the iterative solution indeed converges to the desired approximate solution .section 5 is devoted to the study of approximate problems for lbcf , in a manner similar to section 4 .however , it should be noted that unlike in the case of sbcf , we have to explicitly deal with an additive constant for the pressure . as a result , sometimes there exist multiple solutions for the pressure , especially its additive constant ; other times it is uniquely determined . moreover , in an error analysis , we can only obtain the convergence rate , because of the error of the additive constant of the pressure .if we can eliminate the influence of this error , the same rate - of - convergence as in the case of sbcf is realized .  in section 6 ,several numerical examples are provided to support our theory .we observe that the results of our computation capture the features of sbcf and lbcf and that the numerically calculated errors decrease at for both .section 7 presents the conclusions and discusses some future works .the author learned about ayadi et al . after the completion of the present study .they treat the finite element approximation for the stokes equations with sbcf , using the p1 bubble / p1 element .some numerical examples are presented , and an error estimate is announced without a proof .let be a _ polygonal _ domain in . throughout this paper , we are concerned with the stokes equations written in a familiar form where is the viscosity constant ; , the velocity field ; , the pressure ; and , the external force . as for the boundary , we assume that is a union of two non - overlapping parts , that is , where are relatively nonempty open subsets of . moreover , is assumed to coincide with _ whole one side _ of the polygon for the sake of simplicity .two endpoints of the line segment are respectively denoted by and ; the meaning of these subscripts is clarified in section [ sec3.1 ] .we impose the adhesive boundary condition on , namely , whereas on , we impose one ( and only one ) of the following boundary conditions of friction type : called the _ slip boundary condition of friction type _ ( sbcf ) , and called the _ leak boundary condition of friction type _ ( lbcf ) .the function , called the _ modulus of friction _ , is assumed to be continuous on and strictly positive on .  here , the definitions of the symbols appearing above are as follows :   \(i ) and are constant vectors because is a segment .\(ii ) does not depend on , which is verified by a simple calculation .\(iii ) in ( [ 2.1 ] ) and ( [ 2.2 ] ) , acts as the threshold of the tangential and normal stress beyond which non - trivial slip and leak on may occur , respectively .this is why the boundary conditions ( [ 2.1 ] ) and ( [ 2.2 ] ) are said to be frictional . "we use the usual lebesgue spaces and sobolev spaces for a nonnegative integer , together with their standard norms and semi - norms ( for a space of vector - valued functions , we write , and so on ) . is understood as , and denotes the closure of in .we put and  is also defined for by the norm where is a multi - index and , where , .we also use the sobolev space defined on the boundary for . is understood as , and we put where denotes the surface element of .the usual trace operator defined on onto is denoted by for ; however , we simply write instead of when there is no ambiguity .since and are constant vectors , we immediately obtain the following :  [ lem2.3 ] let . forevery satisfying on and  on , it holds that   in addition , we require the so - called lions - magene space ( see ( * ? ? ? * section i.11 ) ) with its norm defined by where is the distance from to the extreme points of along . we use this space for only one purpose described in the following lemma .[ lem2.1 ] the trace operator maps onto .  see ( * ? ? ?* theorem 1.5.2.3 ) .the lemma implies that the extension to by zero of an arbitrary function in belongs to .now we let and introduce the following two closed subspaces of : which corresponds to the velocity space for sbcf and lbcf , respectively . combining the above two lemmas with the usual trace theorem ,we see that  [ lem2.2 ] ( i ) for every  , it holds that with the constant independent of .\(ii ) every admits an extension  such that with the constant independent of .let us introduce the bilinear forms and are continuous with their operator norms and , respectively , being bounded . as a readily obtainable consequence of korn s inequality ( ( * ? ? ?* lemma 6.2 ) ) , there exists a constant such that this implies that is coercive on and .we simply write and to express and , respectively .then , and , called the barrier terms of friction , are continuous functional on because is bounded on .   for all with satisfying , we obtain green s formula as follows : where the stress vector is defined in section [ sec2.1 ] .in fact , the line integral over appearing in the right - hand side is well defined because . however ,if we have only a lower regularity , say , then the definition of in section [ sec2.1 ] becomes ambiguous .we thus propose a redefinition of as a functional on as follows .[ def2.1 ] let with .when is represented by in the distribution sense , that is , we define by here and hereafter , for a banach space , we denote the dual space of by and the duality pairing between and by .  the functional is well defined according to the trace theorem and the fact that the right - hand side of ( [ 2.8 ] ) vanishes if on , i.e. , .in addition , this definition of agrees with the previous one if and are sufficiently smooth to belong to with .in particular , we see that and are characterized in by respectively , in view of lemma [ lem2.2 ] .herein we assume and with on .with defined by ( [ 2.16 ] ) and , we introduce a weak formulation of the stokes equations ( [ 2.9 ] ) under ( [ 2.10 ] ) and ( [ 2.1 ] ) as follows .find such that is well defined and the slip boundary condition of friction type is satisfied , that is ,  a(u , v ) + b(v , p ) - ( _ , v_)_l^2(_1 ) = ( f , v)_l^2()^2 & , [ 2.4 ] + b(u , q ) = 0 & , + _/gl^(_1 ) |_|g _ 1 , [ 2.14 ] + _u_+ g|u_| = 0 _ 1 .[ 2.12 ]  note that follows from , and thus makes sense .  another formulation by a variational inequality proposed in is  find such that  a(u , v - u ) + b(v - u , p ) + j(v _ ) - j(u _ ) ( f , v - u)_l^2()^2 & , [ 2.3 ] + b(u , q ) = 0 & .the following theorem concerning the existence and uniqueness is essentially derived from ( * ? ? ?* theorems 2.12.3 ) .[ thm2.1 ] ( i ) problems pde and vi are equivalent in the sense that solves problem pde if and only if it solves problem vi .\(ii ) problem vi has a unique solution .  in , another definition of is employed and it is supposed that is smooth , with . however , some slight modification , which is not essential , makes the proofs in applicable to our own situation .   as in the previous subsection , using defined by ( [ 2.17 ] ) and , we introduce a weak formulation of the stokes equations ( [ 2.9 ] ) under ( [ 2.10 ] ) and ( [ 2.2 ] ) as follows .find such that is well defined and the leak boundary condition of friction type is satisfied , that is ,  a(u , v ) + b(v , p ) - ( _ n , v_n)_l^2(_1 ) = ( f , v)_l^2()^2 & , [ 2.13 ] + b(u , q ) = 0 & , + _ n / gl^(_1 ) |_n|g _ 1 , [ 2.15 ] + _ n u_n + g|u_n| = 0 _ 1 .note that follows from , and thus makes sense .another formulation by a variational inequality proposed in is  find such that  a(u , v - u ) + b(v - u , p ) + j(v_n ) - j(u_n ) ( f , v - u)_l^2()^2 & , + b(u , q ) = 0 & .we recall the existence and ( non)uniqueness theorem derived from ( * ? ? ?* theorems 3.13.3 and remark 3.2 ) .[ thm2.2 ] ( i ) problems pde and vi are equivalent in the sense that solves problem pde if and only if it solves problem vi .\(ii ) problem vi has at least one solution , the velocity part of which is unique .\(iii ) if and are two solutions of problem vi , problem pde , there exists a unique constant such that   \(iv ) under the assumptions in ( iii ) , if we suppose on , then .namely , a solution of problem vi is unique .although the definition of and the hypotheses on the boundary in are apparently different from ours , we can complete the proof with only a non - essential modification of the original one in .let be a sequence of triangulations of a polygon , where denotes the length of the greatest side . as usual , we assume that  * is a side , a node , or for all .* and the boundary vertices belong to . * when tends to , each triangle in contains a circle of radius and it is contained in a circle of radius for some constants independent of . *each triangle has at least one vertex that is not on .the one - dimensional meshes of and inherited from the triangulation are denoted respectively by and . for the sets of nodes , we use where the subscripts of s are numbered such that  * s , for , are all vertices of triangles in , which are located in and are arranged in ascending order along .* is the midpoint of and for .in particular , .we denote each side with endpoints by  ] , take two points denoted by and , whose meaning is understood naturally , for .let us define a piecewise constant function on by } + g_{i+\frac12}\eta_{h , i+\frac12}\chi_{[m_{i+\frac16},m_{i+\frac56 } ] } + g_{i+1}\eta_{h , i+1}\chi_{[m_{i+\frac56},m_{i+1 } ] } \big\ } , \label{hhhhhh }   \end{aligned}\ ] ] where denotes the characteristic function of .  then we have by direct computation , it follows that here we have used the inequality to derive the third line .we conclude ( [ 3.45 ] ) from ( [ 3.51 ] ) and ( [ 3.52 ] ) .the estimate ( [ 3.35 ] ) follows similarly if we remark that   \(ii ) let .first , from the proof of ( i ) , we see that   before giving an estimate of which involves , it should be noted that if we have so that in view of the taylor expansion of , we apply ( [ bbbbbb ] ) to deduce )}^2 \notag \\       & \le ch^2\|\eta_h\|_{h^1(e_i)}^2 , \label{dddddd }   \end{aligned}\ ] ] for . by a similar discussion , we have for each .therefore , it follows from ( [ cccccc ] ) and ( [ dddddd])([eeeeee ] ) that so that as a consequence of ( [ ffffff ] ) and ( [ gggggg ] ) , we obtain the desired inequality ( [ 3.21 ] ) by hilbertian interpolation ( see ( * ? ? ? * chapter 14 ) ) between and .as will be shown in theorems [ thm3.2 ] and [ thm4.3 ] below , the leading term of the error is that between and , which is estimated by ( [ 3.21 ] ) with .however , under some additional conditions , we can obtain a sharper estimate than ( [ 3.21 ] ) .[ def3.2 ] an element is said to _ have a constant sign on every side _if , for any , either of the following conditions is satisfied :   .let have a constant sign on every side .if on and on for some , then .[ lem3.5 ] let .if has a constant sign on every side , then moreover , if is a polynomial of degree , then is exact , that is ,   let have a constant sign on every side .because or on for each and is positive on , we have where is defined as ( [ hhhhhh ] ) . summing up these terms, we obtain consequently , it follows that let denote the linear lagrange interpolation of using the nodes in .namely , is continuous on and affine on each side $ ] , satisfying for .then the taylor expansion of implies   now , let us estimate each term appearing in the summation on the right - hand side of ( [ iiiiii ] ) by since simpson s formula is exact for cubic polynomials , we can express thus , due to ( [ kkkkkk ] ) , the first term of ( [ jjjjjj ] ) is bounded from above by note that there holds ( cf .( [ 3.46 ] ) ) for .then , the sum of the first term of ( [ jjjjjj ] ) is estimated as next , the second term of ( [ jjjjjj ] ) is estimated by , which gives hence we conclude from ( [ iiiiii ] ) , ( [ llllll ] ) , and ( [ mmmmmm ] ) that   if is a polynomial of degree , then both terms of ( [ jjjjjj ] ) vanish because , from which ( [ oooooo ] ) follows .this completes the proof .we propose approximate problems for problem vi ( therefore , problem pde ) in the case of sbcf as follows .find such that  a(u_h , v_h - u_h ) + b(v_h - u_h , p_h ) + j_h(v_h ) - j_h(u_h ) ( f , v_h - u_h)_l^2()^2 & , [ 3.15 ] + b(u_h , q_h ) = 0 & .find such that   find such that  a(u_h , v_h ) + b(v_h , p_h ) + ( v_h , _ h)__h = ( f , v_h)_l^2()^2 & , [ 3.13 ] + b(u_h , q_h ) = 0 & , [ 3.11 ] + ( u_h,_h - _ h)__h 0 & .[ 3.12 ]  recall that we are assuming and .we first establish the existence and uniqueness of these approximate problems .[ thm3.1 ] ( i ) problem admits a unique solution .furthermore , it satisfies the following equation :   \(ii ) problems , , and are equivalent in the following sense .\(a ) if is a solution of problem , then there exists a unique such that solves problem .\(b ) if is a solution of problem vi , then there exists a unique such that solves problem ve .\(c ) if is a solution of problem ve , then solves problem vi .\(i ) since the bilinear form is coercive on and the functional is convex , proper , and lower semi - continuous ( actually , continuous ) with respect to the weak topology , we can apply to problem vi a classical existence and uniqueness theorem for second - order elliptic variational inequalities ( see ( * ? ? ?* theorem i.4.1 ) ) .thus , there exists a unique such that ( [ 3.7 ] ) holds .the equation ( [ 3.8 ] ) follows from ( [ 3.7 ] ) with and .\(ii ) ( a ) let be a solution of problem vi .taking as a test function in ( [ 3.7 ] ) , with an arbitrary , we obtain moreover , from lemma [ lem3.1](i ) , we deduce the unique existence of such that by a standard argument .now we let be arbitrary .it follows from lemma [ lem3.2 ] ( ii ) that there exists some such that on , which implies since , we conclude from ( [ 3.7 ] ) , ( [ 3.9 ] ) , and ( [ 3.10 ] ) that hence is a solution of vi .\(b ) let be a solution of vi .taking as a test function in ( [ 3.15 ] ) , with an arbitrary , we have therefore , since , the inf - sup condition given in lemma [ lem4.2 ] asserts the unique existence of such that combining ( [ 3.16 ] ) with ( [ 3.15 ] ) , we obtain which gives , by a triangle inequality , that from ( [ 3.18 ] ) together with lemma [ lem3.2](i ) , we deduce hence lemma [ lem3.3](iii ) implies that , and ( [ 3.13 ] ) is established .it remains only to prove ( [ 3.12 ] ) .taking in ( [ 3.17 ] ) , we have .this implies ( [ 3.12 ] ) by lemma [ lem3.3](ii ) .therefore , is a solution of problem ve .\(c ) let be a solution of problem ve .then we see that from ( [ 3.11 ] ) , and that from ( [ 3.12 ] ) combined with lemma [ lem3.3](ii ) .it follows from ( [ 3.13 ] ) and ( [ 3.14 ] ) that where the last inequality holds by lemma [ lem3.3](i ) .hence is a solution of problem vi .this completes the proof of theorem [ thm3.1 ] .   before presenting the rate - of - convergence results, we state the following :  [ prop3.1 ] let be the solution of problem vi and , that of problem vi for .then ,  \(i ) it holds that   \(ii ) for every and , it holds that  \(iii ) for every , it holds that   \(i ) since is the solution of problem vi by theorem [ thm3.1](ii ) , it satisfies ( [ 3.8 ] ) .hence korn s inequality ( [ 2.5 ] ) , together with the positiveness of , gives which implies ( [ 3.20 ] ) .\(ii ) let and be arbitrary .we begin with the following equality : we bound from above the second term of the right - hand side by ( [ 2.3 ] ) with , the third one by ( [ 3.15 ] ) with itself , and rewrite the fourth one by ( [ 2.4 ] ) with .consequently , combining this with korn s inequality ( [ 2.5 ] ) , we conclude ( [ 3.22 ] ) .\(iii ) taking as a test function in ( [ 2.3 ] ) , with an arbitrary , gives on the other hand we know that ( [ 3.19 ] ) holds , and therefore , by subtraction we obtain now let .it is clear that by lemma [ lem3.1](i ) together with ( [ 3.23 ] ) , we have the desired inequality ( [ 3.26 ] ) follows from ( [ 3.24 ] ) and ( [ 3.25 ] ) .we are now in a position to state the primary result of our error estimates , assuming only the regularity of the exact solution .[ thm3.2 ] let be the solution of problem vi and be that of problem vi for .suppose and with .then we have   we recall the interpolation error estimates ( [ 3.41])([3.42 ] ) .taking in ( [ 3.22 ] ) and ( [ 3.26 ] ) , we find that and that each term of the right - hand side in ( [ 3.27 ] ) is estimated as follows :  \1 .  \2 .  \3 . from ( [ 3.28 ] ) ,   \4 .   \5 . by lemma [ lem3.4](ii ) together with proposition [ prop3.1](i ) ,   \6 .since , lemma [ lem3.4](ii ) implies   \7 . combining these seven estimates with ( [ 3.27 ] ) , we deduce that therefore , we conclude ( [ 3.30 ] ) from ( [ 3.28 ] ) and ( [ 3.29 ] ) and this completes the proof .the previous theorem reveals that the rate of convergence is at best even when the solution is sufficiently smooth .however , it can be improved if additional conditions about the signs of and on are available . to formulate the result, we make the following assumptions ( recall definition [ def3.2 ] and see remark [ rem4.1 ] ) :  ( s1) has a constant sign on every side .( s2) has a constant sign on every side .( s3) on .[ thm3.3 ] in addition to the hypotheses in theorem , we assume and that are satisfied. then we have moreover , if is a polynomial function of degree , we have   we first verify that ( s3 ) implies in fact , for each side , if vanishes on a subset of containing more than three points , then the quadratic polynomial vanishes on the whole .otherwise , we have a.e . on ; hence we deduce from ( [ 2.12 ] ) , namely , that a.e .on . in both cases, it follows that a.e . on .thus ( [ 3.31 ] ) is valid .it follows from ( [ 3.31 ] ) and ( [ 3.32 ] ) that therefore , taking and in ( [ 3.22 ] ) gives let us give estimates for each term on the right - hand side .we can evaluate the first three terms by the same way as in the proof of theorem [ thm3.2 ] . by lemma [ lem3.5 ] ,the fourth and fifth terms are estimated as consequently , we obtain which leads to the estimate for is similar to the proof of theorem [ thm3.2 ] , and then , ( [ 3.48 ] ) follows .  finally ,if is affine then the fourth and fifth terms in ( [ 3.33 ] ) vanish exactly , according to lemma [ lem3.5 ] .hence we have instead of ( [ 3.50 ] ) , from which ( [ 3.49 ] ) follows .[ rem4.1 ] conditions ( s1)(s3 ) are not so artificial .assume that , the velocity part of the solution , is continuous on and that the isolated zeros of on are contained in .if we make sufficiently small , then we see that ( s1 ) and ( s3 ) are satisfied. therefore , since theorem [ thm3.2 ] implies in , we can expect ( s2 ) to also be valid ; however , its rigorous proof is not easy .we propose the following uzawa - type method to compute the solution of problem ve ( therefore , problem vi ) numerically .[ alg3.1 ] choose an arbitrary and . iterate the following two steps for   * * with known , determine by  a(u_h^(k),v_h ) + b(v_h , p_h^(k ) ) = ( f , v_h)_l^2()^2 - ( v_h,_h^(k))__h & , [ 3.34 ] + b(u_h^(k),q_h ) = 0 & . [ 3.54 ]  * * renew by   \(i ) the unique existence of satisfying ( [ 3.34 ] ) and ( [ 3.54 ] ) is guaranteed by the inf - sup condition mentioned in remark [ rem3.1 ] .\(ii ) we can regard ( [ 3.58 ] ) as an approximation of which is equivalent to ( [ 3.12 ] ) by lemma [ lem3.3](ii ) .[ thm3.4 ] let be the solution of problem .under the same notation as algorithm , there exists a constant independent of such that if satisfies , then the iterative solution converges to in , as .  subtracting ( [ 3.34 ] ) from ( [ 3.13 ] ) with test functions in ,we obtain in particular , we take and apply korn s inequality ( [ 2.5 ] ) to obtain next , we note that given in ( [ 3.55 ] ) satisfies as a result of a general property of a projection operator .it follows from ( [ 3.56 ] ) with and , ( [ 3.58 ] ) , ( [ 3.59 ] ) , and ( [ 3.60 ] ) that therefore , since in view of lemmas [ lem3.4](i ) and [ 2.2](i ) , we obtain and thus   on the other hand , by virtue of lemma [ lem3.2](i)(ii ) , we can choose such that and where the constant concerns the equivalence of the norms on the finite dimensional space .hence , it follows from ( [ 3.38 ] ) with that so that since the constant in ( [ 3.37 ] ) is independent of ( and even of ) , if we choose then it follows from ( [ 3.37 ] ) and ( [ 3.39 ] ) that where we may assume ( if not , take ) .consequently , we conclude then , from ( [ 3.57 ] ) it also follows that in as .  finally , subtracting ( [ 3.34 ] ) from ( [ 3.13 ] ) with test functions in gives therefore , by lemma [ lem3.1](i ) we have this completes the proof .approximate problems to problem vi ( therefore , problem pde ) in the case of lbcf are as follows .find such that  a(u_h , v_h - u_h ) + b(v_h - u_h , p_h ) + j_h(v_hn ) - j_h(u_hn ) ( f , v_h - u_h)_l^2()^2 & , [ 4.23 ] + b(u_h , q_h ) = 0 & . [ 4.24 ]  find such that   find such that  a(u_h , v_h ) + b(v_h , p_h ) + ( v_hn , _ h)__h = ( f , v_h)_l^2()^2 & , [ 4.13 ] + b(u_h , q_h ) = 0 & , [ 4.14 ] + ( u_hn , _ h - _ h)__h 0 & . [ 4.15 ]  [ thm4.1 ] ( i ) problem vi admits a unique solution .furthermore , it satisfies the following equation :   \(ii ) problems and are equivalent in the following sense .\(a ) if is a solution of problem vi , then there exists a unique such that solves problem ve .\(b ) if is a solution of problem ve , then solves problem vi .  statements ( i ) and ( ii)(a ) are proved by the same way as theorems [ thm3.1](i ) and ( ii)(b ) , respectively .we obtain ( ii)(b ) by a discussion similar to that in the proof of theorem [ thm3.1](ii)(c ) .[ rem4.2 ] it is clear that if is a solution of problem vi , then is that of problem vi . however , the `` converse '' is no longer true .in fact , the pressure , especially its additive constant , need not to be uniquely determined even if the solution of problem vi is given .this situation is quite different from the case of sbcf .the next theorem guarantees the existence of a solution of problem vi .[ thm4.2 ] ( i ) there exists at least one solution of problem ve , and the velocity part is unique .\(ii ) if and are two solutions of problem ve , then there exists a unique such that   \(iii ) under the assumptions in ( ii ) , if we suppose on , then .namely , a solution of problem is unique .\(i ) the uniqueness of the velocity is obvious ; see remark [ rem4.2 ]  to prove the existence , let be the solution of vi .taking as a test function in ( [ 4.8 ] ) , with an arbitrary , we obtain hence , from lemma [ lem3.1](i ) , we deduce the unique existence of such that moreover , noting that , it follows from lemma [ lem3.2](iii ) that there exists a unique satisfying   for every , by lemma [ lem3.2](ii ) we can choose such that hence , taking in ( [ 4.8 ] ) and ( [ 4.9 ] ) , we obtain substituting ( [ 4.12 ] ) and ( [ 4.11 ] ) into ( [ 4.10 ] ) gives we apply hahn - banach s theorem to deduce the existence of some such that therefore , lemma [ lem3.3](ii ) implies that . furthermore ,since satisfies it follows from lemma [ lem3.3](iv ) that there exists some such that thus , from simpson s formula and ( [ 4.9 ] ) , we obtain this establishes ( [ 4.13 ] ) if we define .equation ( [ 4.14 ] ) obviously holds because .it remains to show ( [ 4.15 ] ) , which is equivalent to by lemma [ lem3.3](ii ) .this is indeed obtained from ( [ 4.16 ] ) with and ( [ 4.12 ] ) .\(ii ) let and be two solutions of problem ve . because the uniqueness of the pressure up to additive constants is shown in the proof of ( i ) , there exists a unique constant such that .since and satisfy ( [ 4.13 ] ) , subtracting the two equations and calculating in a manner similar to ( [ 4.17])([4.16 ] ) , we obtain here , is defined by for .it follows from ( [ 4.30 ] ) together with lemma [ lem3.2](i ) that for all .hence , and ( [ 4.31 ] ) is proved .\(iii ) the assumption implies that either of the following is true :  \(a ) there exists such that .\(b ) there exists such that .  since , we see that therefore , both ( a ) and ( b ) above must hold true . then , from ( [ 4.15 ] ) combined with lemma [ lem3.3](ii ) , we find some satisfying and .consequently , the additive constant appearing in ( ii ) can not attain any value except because .this completes the proof of theorem [ thm4.2 ] .let us begin with the following analogue of proposition [ prop3.1 ] .[ prop4.1 ] let be a solution of problem vi and be that of problem vi for .then ,  \(i ) it holds that   \(ii ) for every and , it holds that   \(iii ) for every , it holds that where and .\(iv ) for every , it holds that   statements other than ( iv ) can be proved by the same way as proposition [ prop3.1 ] . to show ( iv ), we let .it is clear that . to bound the latter term, we deduce from lemma [ lem3.1 ] , together with ( [ 2.13 ] ) and ( [ 4.13 ] ) , that here , to derive ( [ 4.18 ] ) , we have used the estimates which are obtained from lemmas [ lem2.2](i ) , [ lem3.3](i ) , and [ lem3.4](i ) .the desired inequality ( [ 4.19 ] ) immediately follows from ( [ 4.18 ] ) .if and , we can take in ( [ 4.19 ] ) according to the equality combined with ( [ 4.20 ] ) .we state the rate of convergence result for the case of lbcf , which is not better than that of sbcf because of the influence of an additive constant of the pressure .[ thm4.3 ] let be a solution of problem vi and be that of problem vi for , and suppose with .then , where and .let us take and in ( [ 4.21 ] ) and bound from above each term on the right - hand side .by ( [ 4.19 ] ) , we have for the other terms , we employ the same estimates as those in the proof of theorem [ thm3.2 ] .then it follows that which implies  follows from ( [ 4.20 ] ) and ( [ 4.22 ] ) and this completes the proof .\(i ) if we assume , in addition , that then we can establish the result of . moreover , as in the case of sbcf , under suitable conditions regarding the signs of and on , it can be improved to , or even if is affine .\(ii ) when the uniqueness of problem vi holds , we can obtain a strong convergence result for the error of the pressure including the additive constant .in fact , the uniform boundedness of in gives a weak convergence limit for some subsequence .since in , we have therefore , taking the limit in ( [ 4.23])([4.24 ] ) , we find that is a solution of problem vi .hence , and from and , we conclude the strong convergence of the whole sequence .based on problem ve , we propose the following uzawa - type method to compute the approximate solution numerically .[ alg4.1 ] choose an arbitrary and . iterate the following two steps for   * * with known , determine by  a(u_h^(k),v_h ) + b(v_h , p_h^(k ) ) = ( f , v_h)_l^2()^2 - ( v_hn , _ h^(k))__h & , [ 4.25 ] + b(u_h^(k),q_h ) = 0 & . [ 4.32 ]  * * renew by   the unique existence of satisfying ( [ 4.25 ] ) and ( [ 4.32 ] ) is guaranteed by the inf - sup condition in lemma [ lem4.2 ] .[ thm4.4 ] under the same notation as algorithm [ alg4.1 ] , there exists a constant independent of such that if satisfies , then converges to some solution of problem in , as .first we show the boundedness of the sequence .in fact , taking in ( [ 4.25 ] ) , we find from korn s inequality ( [ 2.5 ] ) that which gives . here , to derive the last line , we have used which is obtained from lemmas [ lem3.3](i ) , [ lem3.4](i ) , and [ lem2.2](i ). then lemma [ lem4.2 ] , together with ( [ 4.25 ] ) , implies that where in the third line is estimated in a manner similar to ( [ 4.33 ] ) .it is clear that is bounded because .therefore , we can extract a subsequence converging to some element . making and in ( [ 4.25])([4.26 ] ), we obtain  a(u_h , v_h ) + b(v_h , p_h ) + ( v_hn , _ h)__h = ( f , v_h)_l^2()^2 & , + b(u_h , q_h ) = 0 & , + _ h = _ _ h(_h + u_hn ) .consequently , since the last equation is equivalent to ( [ 4.15 ] ) by virtue of lemma [ lem3.3](ii ) , we see that is a solution of problem ve .it remains only to prove that the whole sequence converges to .subtracting ( [ 4.25 ] ) from ( [ 4.13 ] ) , we have in particular , if we take , then therefore , it follows from a general property of the projection operator that where we have applied korn s inequality ( [ 2.5 ] ) and the estimate to derive the last line .since the constant in ( [ 4.29 ] ) is independent of ( and even of ) , we choose to obtain hence the sequence is decreasing . noting that a decreasing sequence in , bounded from below , converges to its infimum and that by construction , we conclude that in as . from this and ( [ 4.27 ] ) , it also follows that in . finally , from the inf - sup condition given in lemma [ lem4.2 ] combined with ( [ 4.28 ] ), we have this completes the proof .\(i ) the resulting solution of problem ve as the limit of , especially its additive constant of the pressure , may depend on a choice of or that of the starting value .however , if on , and hence the uniqueness of the solution of problem ve is valid , then it is obviously independent of them .\(ii ) contrary to the case of sbcf , it is difficult to prove an exponential convergence of the iterative solution because we do not know whether , which is necessary to deduce an extension of to .we assume , the boundary of which consists of two portions and given by in particular , the set of extreme points is . for the triangulation of , we employ a uniform friedrichs - keller type mesh , where denotes the division number of each side of the square .let us consider which turns out to be the solution of the stokes equations under the adhesive boundary condition for and given by  f_1(x , y ) = 0 , + f_2(x , y ) = 120(2x-1)y^2(1-y)^2 + 80x(1-x)(1 - 2x)(6y^2 - 6y+1 ) + 8(6x^5 - 15x^4 + 10x^3 ) .  by direct computation , we have now , if we impose sbcf or lbcf on , with being constant , instead of the adhesive boundary condition , then in the case of sbcf , we find that and in the case of lbcf , we indeed observe some of the abovementioned phenomena in our numerical computation , as indicated in the plots of the velocity field shown below in figures [ fig5.1 ] and [ fig5.2 ] .in addition , we find that the bigger ( resp .smaller ) the threshold of a tangential or normal stress becomes , the more difficult ( resp .easier ) it becomes for a non - trivial slip or leak to occur , which is in agreement with our natural intuition ..values of the lagrange multiplier on [ cols="^,^,^,^,^,^,^,^ " , ]   finally , we evaluate the error between approximate solutions and exact ones as the division number increases , when and for the case of sbcf and lbcf , respectively . since we do not know the explicit exact solutions , we employ the approximate solutions with as the reference solutions , and numerically calculate and . here , the additive constants of s are chosen such that .then , as table [ tab5.2 ] shows , we can observe the optimal order convergence for both sbcf and lbcf .a finite element analysis using the p2/p1 element to the stokes equations under sbcf or lbcf is examined .we have proved the existence and uniqueness ( partial non - uniqueness ) results and established the convergence order as error estimates for appropriately smooth solutions ; sufficient conditions to obtain the optimal order are also presented . to compute the approximate solution ,we have proposed an iterative uzawa - type algorithm .we have applied it to some examples and numerically observed the convergence order of .  in a future study, we would like to extend our theory to a more general situation , for example , a smooth domain without corners , nonlinear navier - stokes equations , a case in which sbcf and lbcf are imposed simultaneously , or a time - dependent problem .i would like to thank dr .hirofumi notsu for providing a crucial idea that helped improve the performance of the numerical experiment .i would also like to thank professors norikazu saito and hiroshi suito for bringing this topic to my attention and encouraging me through valuable discussions .this work was supported by crest , jst .

Abstract: a finite element approximation of the stokes equations under a certain nonlinear boundary condition , namely , the slip or leak boundary condition of friction type , is considered . we propose an approximate problem formulated by a variational inequality , prove an existence and uniqueness result , present an error estimate , and discuss a numerical realization using an iterative uzawa - type method . several numerical examples are provided to support our theoretical results .