Provable stable arbitrary order symmetric interior penalty discontinuous Galerkin (SIP) discretisations of variable viscosity, incompressible Stokes flow utilising @math -- @math elements and hierarchical Legendre basis polynomials are developed and investigated.For solving the resulting linear system, a block preconditioned iterative method is proposed. The nested viscous problem is solved by a @math -multilevel preconditioned Krylov subspace method. For the @math -coarsening, a twolevel method utilising element-block Jacobi preconditioned iterations as a smoother is employed. Piecewise bilinear ( @math ) and piecewise constant ( @math ) @math -coarse spaces are considered. Finally, Galerkin @math -coarsening is proposed and investigated for the two @math -coarse spaces considered. Through a number of numerical experiments, we demonstrate that utilising the @math coarse space results in the most robust @math -multigrid method for variable viscosity Stokes flow. Using this @math coarse space we observe that the convergence of the overall Stokes solver appears to be robust with respect to the jump in the viscosity and only mildly depending on the polynomial order @math . It is demonstrated and supported by theoretical results that the convergence of the SIP discretisations and the iterative methods rely on a sharp choice of the penalty parameter based on local values of the viscosity.
A continuous interior penalty hp-finite element method that penalizes the jump of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for first-order and advection-dominated transport operators. The analysis relies on three technical results that are of independent interest: an hp- inverse trace inequality, a local discontinuous to continuous hp-interpolation result, and hp-error estimates for continuous L2-orthogonal projections. We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.  In this paper we construct Discontinuous Galerkin approximations of the Stokes problem where the velocity field is @math H ( div , Ω ) -conforming. This implies that the velocity solution is divergence-free in the whole domain. This property can be exploited to design a simple and effective preconditioner for the final linear system.  We consider the behavior of the GMRES method for solving a linear system @math when @math is singular or nearly so, i.e., ill conditioned. The (near) singularity of @math may or may not affect the performance of GMRES, depending on the nature of the system and the initial approximate solution. For singular @math , we give conditions under which the GMRES iterates converge safely to a least-squares solution or to the pseudoinverse solution. These results also apply to any residual minimizing Krylov subspace method that is mathematically equivalent to GMRES. A practical procedure is outlined for efficiently and reliably detecting singularity or ill conditioning when it becomes a threat to the performance of GMRES. This paper reviews some known and proposes some new preconditioning methods for a number of discontinuous Galerkin (or DG) finite element approximations for elliptic problems of second order. Nested hierarchy of meshes is generally assumed. Our approach utilizes a general two-level scheme, where the finite element space for the DG method is decomposed into a subspace (viewed as an auxiliary or ‘coarse’ space), plus a correction which can be handled by a standard smoothing procedure. We consider three different auxiliary subspaces, namely, piecewise linear C0-conforming functions, piecewise linear functions that are continuous at the centroids of the edges faces (Crouzeix-Raviart finite elements) and piecewise constant functions over the finite elements. To support the theoretical results, we also present numerical experiments for 3-D model problem showing uniform convergence of the constructed methods. Copyright © 2006 John Wiley & Sons, Ltd. A multigrid method for the Stokes system discretized with an Hdiv-conforming discontinuous Galerkin method is presented. It acts on the combined velocity and pressure spaces and thus does not need a Schur complement approximation. The smoothers used are of overlapping Schwarz type and employ a local Helmholtz decomposition. Additionally, we use the fact that the discretization provides nested divergence free subspaces. We present the convergence analysis and numerical evidence that convergence rates are not only independent of mesh size, but also reasonably small. Abstract The diapiric ascent of light rocks through a denser oberburden is simulated using a new numerical method based in part on finite differences. Rocks are modelled as viscous Newtonian liquids and the internal interfaces are located with the help of passive markers. These markers are relocated using a new algorithm equivalent to a cellular automaton rule. This allows the markers to be redistributed very rapidly. This new method appears to be well adapted to quickly resolving geological problems. Simulations of the growth of a single dome are performed under various viscosity ratios for the salt and the overburden. The effect of asymmetrical initial deformations of the salt layer is to produce diapiric structures that slowly return to symmetry as the diapir matures and rises in an isotropic medium. Simulations involving continuous sedimentation of the overburden are also performed. An initial geometry on a basin margin in which multiple diapirs eventually develop is studied. Diapirs grow relatively rapidly near the centre, while on the margins the salt layer flows upward following the basement. Simulations with salt that is more viscous than sediments show the development of previously unnoticed asymmetrical structures.