Anisotropic Elliptic Problems Research Articles

Corrigendum to Existence and Regularity Results for Anisotropic Elliptic Problems

Corrigendum to Existence and Regularity Results for Anisotropic Elliptic Problems

Overdetermined anisotropic elliptic problems

A symmetry result is established for solutions to overdetermined anisotropic elliptic problems in variational form, which extends Serrin’s theorem dealing with the isotropic radial case. The involved anisotropy arises from replacing the Euclidean norm of the gradient with an arbitrary norm in the associated variational integrals. The resulting symmetry of the solutions is that of the so-called Wulff shape.

Nonoverlapping Schwarz algorithm for solving two-dimensional m-DDFV schemes

We propose a nonoverlapping Schwarz algorithm for solving ‘discrete duality finite-volume’ (DDFV) schemes on general meshes. In order to handle this problem the first step is to propose and study a convenient DDFV scheme for anisotropic elliptic problems with mixed Dirichlet/Fourier boundary conditions. Then we are able to build the corresponding Schwarz algorithm and to prove its convergence to the solution of the DDFV scheme on the initial domain. We finally give some numerical results in the case where the Schwarz iterations are used both as a solver and as a preconditioner.

Existence and Regularity Results for Anisotropic Elliptic Problems

Abstract We study existence and regularity of the solutions for some anisotropic elliptic problems with homogeneous Dirichlet boundary conditions in bounded domains.

Iterative multiscale finite-volume method

The multiscale finite-volume (MSFV) method for the solution of elliptic problems is extended to an efficient iterative algorithm that converges to the fine-scale numerical solution. The localization errors in the MSFV method are systematically reduced by updating the local boundary conditions with global information. This iterative multiscale finite-volume (i-MSFV) method allows the conservative reconstruction of the velocity field after any iteration, and the MSFV method is recovered, if the velocity field is reconstructed after the first iteration. Both the i-MSFV and the MSFV methods lead to substantial computational savings, where an approximate but locally conservative solution of an elliptic problem is required. In contrast to the MSFV method, the i-MSFV method allows a systematic reduction of the error in the multiscale approximation. Line relaxation in each direction is used as an efficient smoother at each iteration. This smoother is essential to obtain convergence in complex, highly anisotropic, heterogeneous domains. Numerical convergence of the method is verified for different test cases ranging from a standard Poisson equation to highly heterogeneous, anisotropic elliptic problems. Finally, to demonstrate the efficiency of the method for multiphase transport in porous media, it is shown that it is sufficient to apply the iterative smoothing procedure for the improvement of the localization assumptions only infrequently, i.e. not every time step. This result is crucial, since it shows that the overall efficiency of the i-MSFV algorithm is comparable with the original MSFV method. At the same time, the solutions are significantly improved, especially for very challenging cases.

Compact Multiscale Finite Volume Method for Heterogeneous Anisotropic Elliptic Equations

The multiscale finite volume (MSFV) method is introduced for the efficient solution of elliptic problems with rough coefficients in the absence of scale separation. The coarse operator of the MSFV method is presented as a multipoint flux approximation (MPFA) with numerical evaluation of the transmissibilities. The monotonicity region of the original MSFV coarse operator has been determined for the homogeneous anisotropic case. For grid-aligned anisotropy the monotonicity of the coarse operator is very limited. A compact coarse operator for the MSFV method is presented that reduces to a 7-point stencil with optimal monotonicity properties in the homogeneous case. For heterogeneous cases the compact coarse operator improves the monotonicity of the MSFV method, especially for anisotropic problems. The compact operator also leads to a coarse linear system much closer to an M-matrix. Gradients in the direction of strong coupling vanish in highly anisotropic elliptic problems with homogeneous Neumann boundary d...

A multilevel method for discontinuous Galerkin approximation of three‐dimensional anisotropic elliptic problems

AbstractWe construct optimal order multilevel preconditioners for interior‐penalty discontinuous Galerkin (DG) finite element discretizations of three‐dimensional (3D) anisotropic elliptic boundary‐value problems. In this paper, we extend the analysis of our approach, introduced earlier for 2D problems (SIAM J. Sci. Comput., accepted), to cover 3D problems. A specific assembling process is proposed, which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two‐level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. A new bound for the constant γ in the strengthened Cauchy–Bunyakowski–Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Copyright © 2007 John Wiley & Sons, Ltd.

An immersed interface method for anisotropic elliptic problems on irregular domains in 2D

AbstractThis work is a generalization of the immersed interface method for discretization of a nondiagonal anisotropic Laplacian in 2D. This first‐order discretization scheme enforces weakly diagonal dominance of the numerical scheme whenever possible. A necessary and sufficient condition depending on the mesh size h for the existence of this scheme at an interior grid point is found in terms of the anisotropy matrix. A linear programming approach is introduced for finding the weights of the schemes. The method is tested with a parametrized family of anisotropic Poisson equations. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Résolution numérique d'un problème elliptique fortement anisotrope en deux dimensions par une méthode de paramétrisation

We introduce a numerical method for solving an anisotropic elliptic problem. We address the case where the direction of the anisotropy varies, and the anisotropy is high. A finite volume scheme is implemented to solve the problem for small anisotropy ratio, then the parameterization method consists in devising an extrapolation of the solution of the anisotropic problem by combining solutions of a sequence of isotropic problems. To cite this article: P. Guillaume, V. Latocha, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.

An Immersed Interface Method for Solving Anisotropic Elliptic Boundary Value Problems in Three Dimensions

We have developed a first-order stable Cartesian grid discretization that uses only interior grid points for inhomogeneous anisotropic elliptic operators subject to Neumann boundary conditions on a bounded nonrectangular geometry in three dimensions. For this discretization method, a necessary and sufficient condition depending on the mesh size h for the existence of this first-order stable scheme at a regular (i.e., interior) grid point is found in terms of the anisotropy matrix. For this discretization method, a way to analyze the existence of a first-order stable scheme at an irregular (i.e., boundary) grid point is also given. The arguments are identical to those for the two-dimensional case [M. A. Dumett and J. P. Keener, A Numerical Method for Solving Anisotropic Elliptic Boundary Value Problems on an Irregular Domain in 2D, manuscript]; only the details change. Unlike in [M. A. Dumett and J. P. Keener, A Numerical Method for Solving Anisotropic Elliptic Boundary Value Problems on an Irregular Domain in 2D, manuscript], a discussion of Dirichlet and Robin boundary conditions is also included. In particular, it is shown that the Gerschgorin condition does not impose sign restrictions on irregular grid points stencil coefficients as in the Neumann case.

Analysis of tensor product multigrid

We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced.

On the Additive Version of the Algebraic Multilevel Iteration Method for Anisotropic Elliptic Problems

In this paper a recently proposed additive version of the algebraic multilevel iteration method for iterative solution of elliptic boundary value problems is studied. The method constructs a nearly optimal order parameter-free preconditioner, which is robust with respect to anisotropy and discontinuity of the problem coefficients. It uses a new strategy for approximating the blocks corresponding to "new" basis functions on each discretization level. To cope with the difficulties arising from the anisotropy, the problem on the coarsest mesh is solved using a bordering technique with a special choice of bordering vectors. The aim is to find a parameter-free "black-box" robust solver. The results are derived in the framework of a hierarchical basis, linear finite element discretization of an elliptic problem on arbitrary triangular meshes, and a hierarchical basis, bilinear finite element discretization on Cartesian meshes. A comparison of the method with some other iterative solution techniques is presented. Robustness and high efficiency of the proposed algorithm are demonstrated on several model-type problems.

Circulant block-factorization preconditioning of anisotropic elliptic problems

The recently introduced circulant block-factorization preconditioners are studied in this paper. The general approach is first formulated for the case of block-tridiagonal sparse matrices. Then an estimate of the condition number of the preconditioned matrix for a model anisotropic Dirichlet boundary value problem is derived in the formκ<√2e(n+1)+2, whereN=n 2 is the size of the discrete problem, ande stands for the ratio of the anisotropy. Various numerical tests demonstrating the behavior of the circulant block-factorization preconditioners for anisotropic problems are presented.

Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems

We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in ℝ d . Our analysis is based on the theory of additive subspace correction methods and applies to finite element and prewavelet schemes. We present multilevel- and prewavelet-based methods that are robust for anisotropic diffusion operators with additional Helmholtz term. Furthermore, the resulting convergence rates are independent of the discretization level. Beside their theoretical foundation, we also report on the results of various numerical experiments to compare the different methods.

Algebraic Multilevel Preconditioning of Anisotropic Elliptic Problems

In this paper the recently proposed algebraic multilevel iteration method for iterative solution of elliptic boundary value problems with anisotropy and discontinuous coefficients is studied. Based on a special approximation of the blocks corresponding to the new nodes at every discretization level, an optimal order preconditioner with respect to the arithmetic cost independent of both the discontinuity and the anisotropy of the coefficients is constructed. The advantages of the proposed algorithms are illustrated by numerical tests.

Fourier Analysis of Incomplete Factorization Preconditioners for Three-Dimensional Anisotropic Problems

To solve three-dimensional elliptic problems using preconditioned conjugate gradient, it is crucial to make a good choice of preconditioner. To facilitate this choice, a Fourier analysis technique has been used by Chan and Elman [SIAM Rev., 31 (1989), pp. 20–49.] and others to study preconditioned systems arising from the discretization of the two-dimensional model elliptic equation. In this paper the same technique is used to analyze relaxed-modified incomplete factorization preconditioned systems that arise from the discretization of a three-dimensional anisotropic elliptic problem. Expressions for the “Fourier eigenvalues” of the preconditioned three-dimensional systems are presented along with estimates of the condition numbers. For MILU, an optimal value for the parameter c is derived. The correlation between the distribution of the eigenvalues and the Fourier results for the preconditioned systems is remarkable. From the expressions for the eigenvalues we prove that $\kappa (M^{ - 1} A)$ is order $h^{ - 2} $ for ILU and order $h^{ - 1} $ for $\operatorname{MILU}(c \ne 0)$. Then by examining the distribution of Fourier eigenvalues, the dependence of PCG convergence rate on the clustering of the eigenvalues of an operator, as well as its condition number, can be exemplified. The PCG experiments were performed on an Alliant FX/80.