Variational multiscale method

The variational multiscale method (VMS) is a technique used for deriving models and numerical methods for multiscale phenomena. The VMS framework has been mainly applied to design stabilized finite element methods in which stability of the standard Galerkin method is not ensured both in terms of singular perturbation and of compatibility conditions with the finite element spaces.
Stabilized methods are getting increasing attention in computational fluid dynamics because they are designed to solve drawbacks typical of the standard Galerkin method: advection-dominated flows problems and problems in which an arbitrary combination of interpolation functions may yield to unstable discretized formulations. The milestone of stabilized methods for this class of problems can be considered the Streamline Upwind Petrov-Galerkin method (SUPG), designed during 80s for convection dominated-flows for the incompressible Navier–Stokes equations by Brooks and Hughes. Variational Multiscale Method (VMS) was introduced by Hughes in 1995. Broadly speaking, VMS is a technique used to get mathematical models and numerical methods which are able to catch multiscale phenomena; in fact, it is usually adopted for problems with huge scale ranges, which are separated into a number of scale groups. The main idea of the method is to design a sum decomposition of the solution as formula_1, where formula_2 is denoted as coarse-scale solution and it is solved numerically, whereas formula_3 represents the fine scale solution and is determined analytically eliminating it from the problem of the coarse scale equation.
The abstract framework.
Abstract Dirichlet problem with variational formulation.
Consider an open bounded domain formula_4 with smooth boundary formula_5, being formula_6 the number of space dimensions. Denoting with formula_7 a generic, second order, nonsymmetric differential operator, consider the following boundary value problem: 
being formula_10 and formula_11 given functions. Let formula_12 be the Hilbert space of square-integrable functions with square-integrable derivatives: 
Consider the trial solution space formula_14 and the weighting function space formula_15 defined as follows:
The variational formulation of the boundary value problem defined above reads:
being formula_19 the bilinear form satisfying formula_20, formula_21 a bounded linear functional on formula_15 and formula_23 is the formula_24 inner product. Furthermore, the dual operator formula_25 of formula_7 is defined as that differential operator such that formula_27.
Variational multiscale method.
In VMS approach, the function spaces are decomposed through a multiscale direct sum decomposition for both formula_28 and formula_29 into coarse and fine scales subspaces as:
and
Hence, an "overlapping" sum decomposition is assumed for both formula_32 and formula_33 as:
where formula_2 represents the "coarse" (resolvable) scales and formula_3 the "fine" (subgrid) scales, with formula_37, formula_38, formula_39 and formula_40. In particular, the following assumptions are made on these functions:
With this in mind, the variational form can be rewritten as
and, by using bilinearity of formula_43 and linearity of formula_44,
Last equation, yields to a coarse scale and a fine scale problem:
or, equivalently, considering that formula_20 and formula_21:
By rearranging the second problem as formula_52, the corresponding Euler–Lagrange equation reads:
which shows that the fine scale solution formula_3 depends on the strong residual of the coarse scale equation formula_55. The fine scale solution can be expressed in terms of formula_55 through the Green's function formula_57:
Let formula_59 be the Dirac delta function, by definition, the Green's function is found by solving formula_60
Moreover, it is possible to express formula_3 in terms of a new differential operator formula_63 that approximates the differential operator formula_64 as 
formula_65
with formula_66. In order to eliminate the explicit dependence in the coarse scale equation of the sub-grid scale terms, considering the definition of the dual operator, the last expression can be substituted in the second term of the coarse scale equation: 
Since formula_63 is an approximation of formula_64, the Variational Multiscale Formulation will consist in finding an approximate solution formula_70 instead of formula_2. The coarse problem is therefore rewritten as: 
being 
Introducing the form 
and the functional 
the VMS formulation of the coarse scale equation is rearranged as: 
Since commonly it is not possible to determine both formula_63 and formula_78, one usually adopt an approximation. In this sense, the coarse scale spaces formula_79 and formula_80 are chosen as finite dimensional space of functions as:
and
being formula_83 the Finite Element space of Lagrangian polynomials of degree formula_84 over the mesh built in formula_85 . Note that formula_86 and formula_87 are infinite-dimensional spaces, while formula_88 and formula_89 are finite-dimensional spaces.
Let formula_90 and formula_91 be respectively approximations of formula_92 and formula_93, and let formula_94 and formula_95 be respectively approximations of formula_96 and formula_97. The VMS problem with Finite Element approximation reads: 
or, equivalently:
VMS and stabilized methods.
Consider an advection–diffusion problem: 
where formula_101 is the diffusion coefficient with formula_102 and formula_103 is a given advection field. Let formula_104 and formula_105, formula_106, formula_107. Let formula_108, being formula_109 and formula_110.
The variational form of the problem above reads: 
being
Consider a Finite Element approximation in space of the problem above by introducing the space formula_113 over a grid formula_114 made of formula_115 elements, with formula_116.
The standard Galerkin formulation of this problem reads
Consider a strongly consistent stabilization method of the problem above in a finite element framework: 
for a suitable form formula_119 that satisfies: 
The form formula_119 can be expressed as formula_122, being formula_123 a differential operator such as:
and formula_125 is the stabilization parameter. A stabilized method with formula_126 is typically referred to multiscale stabilized method . In 1995, Thomas J.R. Hughes showed that a stabilized method of multiscale type can be viewed as a sub-grid scale model where the stabilization parameter is equal to 
or, in terms of the Green's function as
which yields the following definition of formula_125: 
VMS turbulence modeling for large-eddy simulations of incompressible flows.
The idea of VMS turbulence modeling for Large Eddy Simulations(LES) of incompressible Navier–Stokes equations was introduced by Hughes et al. in 2000 and the main idea was to use - instead of classical filtered techniques - variational projections.
Incompressible Navier–Stokes equations.
Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density formula_131 in a domain formula_132 with boundary formula_133, being formula_134 and formula_135 portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied (formula_136):
being formula_138 the fluid velocity, formula_139 the fluid pressure, formula_140 a given forcing term, formula_141 the outward directed unit normal vector to formula_135, and formula_143 the viscous stress tensor defined as: 
Let formula_145 be the dynamic viscosity of the fluid, formula_146 the second order identity tensor and formula_147 the strain-rate tensor defined as: 
The functions formula_149 and formula_150 are given Dirichlet and Neumann boundary data, while formula_151 is the initial condition.
Global space time variational formulation.
In order to find a variational formulation of the Navier–Stokes equations, consider the following infinite-dimensional spaces: 
Furthermore, let formula_155 and formula_156. The weak form of the unsteady-incompressible Navier–Stokes equations reads: given formula_157,
where formula_23 represents the formula_24 inner product and formula_162 the formula_163 inner product. Moreover, the bilinear forms formula_164, formula_165 and the trilinear form formula_166 are defined as follows:
Finite element method for space discretization and VMS-LES modeling.
In order to discretize in space the Navier–Stokes equations, consider the function space of finite element
of piecewise Lagrangian Polynomials of degree formula_84 over the domain formula_85 triangulated with a mesh formula_171 made of tetrahedrons of diameters formula_172, formula_173. 
Following the approach shown above, let introduce a multiscale direct-sum decomposition of the space formula_174 which represents either formula_175 and formula_176:
being 
the finite dimensional function space associated to the coarse scale, and 
the infinite-dimensional fine scale function space, with 
and 
An overlapping sum decomposition is then defined as:
By using the decomposition above in the variational form of the Navier–Stokes equations, one gets a coarse and a fine scale equation; the fine scale terms appearing in the coarse scale equation are integrated by parts and the fine scale variables are modeled as:
In the expressions above, formula_185 and formula_186 are the residuals of the momentum equation and continuity equation in strong forms defined as: 
while the stabilization parameters are set equal to:
where formula_189 is a constant depending on the polynomials's degree formula_190, formula_191 is a constant equal to the order of the backward differentiation formula (BDF) adopted as temporal integration scheme and formula_192 is the time step. The semi-discrete variational multiscale multiscale formulation (VMS-LES) of the incompressible Navier–Stokes equations, reads: given formula_157,
being 
and
The forms formula_197 and formula_198 are defined as: 
From the expressions above, one can see that: 