Maximum principle for the finite element solution of time‐dependent anisotropic diffusion problems
Copyright policy )Maximum principle for the finite element solution of time‐dependent anisotropic diffusion problems
AbstractPreservation of the maximum principle is studied for the combination of the linear finite element method in space and the θ ‐method in time for solving time‐dependent anisotropic diffusion problems. It is shown that the numerical solution satisfies a discrete maximum principle when all element angles of the mesh measured in the metric specified by the inverse of the diffusion matrix are nonobtuse, and the time step size is bounded below and above by bounds proportional essentially to the square of the maximal element diameter. The lower bound requirement can be removed when a lumped mass matrix is used. In two dimensions, the mesh and time step conditions can be replaced by weaker Delaunay‐type conditions. Numerical results are presented to verify the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013
- University of Kansas United States
- University of Kansas Center for Research Inc United States
- University of Central Arkansas United States
\(\theta\)-method, anisotropic diffusion, Numerical Analysis (math.NA), stepsize bound, Maximum principles in context of PDEs, time-dependent, maximum principle, linear finite element method, numerical result, finite element, Initial-boundary value problems for second-order parabolic equations, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Mathematics - Numerical Analysis, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
\(\theta\)-method, anisotropic diffusion, Numerical Analysis (math.NA), stepsize bound, Maximum principles in context of PDEs, time-dependent, maximum principle, linear finite element method, numerical result, finite element, Initial-boundary value problems for second-order parabolic equations, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Mathematics - Numerical Analysis, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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