A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems

Nonlinear constrained finite element approximations to anisotropic diffusion problems are considered. Starting with a standard (linear or bilinear) Galerkin discretization, the entries of the stiffness matrix are adjusted so as to enforce sufficient conditions of the discrete maximum principle (DMP). An algebraic splitting is employed to separate the contributions of negative and positive off-diagonal coefficients which are associated with diffusive and antidiffusive numerical fluxes, respectively. In order to prevent the formation of spurious undershoots and overshoots, a symmetric slope limiter is designed for the antidiffusive part. The corresponding upper and lower bounds are defined using an estimate of the steepest gradient in terms of the maximum and minimum solution values at surrounding nodes. The recovery of nodal gradients is performed by means of a lumped-mass L"2 projection. The proposed slope limiting strategy preserves the consistency of the underlying discrete problem and the structure of the stiffness matrix (symmetry, zero row and column sums). A positivity-preserving defect correction scheme is devised for the nonlinear algebraic system to be solved. Numerical results and a grid convergence study are presented for a number of anisotropic diffusion problems in two space dimensions.

[1]  Hans D. Mittelmann,et al.  Some remarks on the discrete maximum-principle for finite elements of higher order , 1981, Computing.

[2]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[3]  Sergey Korotov,et al.  Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions , 2005, Numerische Mathematik.

[4]  Sergey Korotov,et al.  On discrete maximum principles for nonlinear elliptic problems , 2007, Math. Comput. Simul..

[5]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE , 1995 .

[6]  Christophe Le Potier,et al.  Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structurés , 2005 .

[7]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[8]  Richard Liska,et al.  Enforcing the Discrete Maximum Principle for Linear Finite Element Solutions of Second-Order Elliptic Problems , 2007 .

[9]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[10]  D. Kuzmin,et al.  On the design of general-purpose flux limiters for implicit FEM with a consistent mass matrix , 2005 .

[11]  Ming Xue,et al.  High-Order Monotonic Numerical Diffusion and Smoothing , 2000 .

[12]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[13]  Róbert Horváth,et al.  Discrete maximum principle for linear parabolic problems solved on hybrid meshes , 2005 .

[14]  Dmitri Kuzmin,et al.  On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection , 2006, J. Comput. Phys..

[15]  Eric Deleersnijder,et al.  Numerical discretization of rotated diffusion operators in ocean models , 2000 .

[16]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[17]  Ramon Codina,et al.  A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation , 1993 .

[18]  Willem Hundsdorfer,et al.  A note on flux limiting for diffusion discretizations , 2003 .

[19]  Dmitri Kuzmin,et al.  Adaptive mesh refinement for high‐resolution finite element schemes , 2006 .

[20]  Stefan Turek,et al.  Flux-corrected transport : principles, algorithms, and applications , 2005 .

[21]  Gisbert Stoyan,et al.  On maximum principles for monotone matrices , 1986 .

[22]  Dmitri Kuzmin,et al.  Algebraic Flux Correction I. Scalar Conservation Laws , 2005 .

[23]  Philippe G. Ciarlet,et al.  Discrete maximum principle for finite-difference operators , 1970 .

[24]  Chia-Jung Hsu.  Numerical Heat Transfer and Fluid Flow , 1981 .

[25]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[26]  P. G. Ciarlet,et al.  Maximum principle and uniform convergence for the finite element method , 1973 .

[27]  Zhiqiang Sheng,et al.  Monotone finite volume schemes for diffusion equations on polygonal meshes , 2008, J. Comput. Phys..