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Stability of Explicit One-Step Methods for P1-Finite Element Approximation of Linear Diffusion Equations on Anisotropic Meshes
Huang, Weizhang Kamenski, Lennard Lang, Jens
Huang, Weizhang
Kamenski, Lennard
Lang, Jens
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Abstract
We study the stability of explicit one-step integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of $2(d+1)$, where $d$ is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The second factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.
Description
The research of the authors was supported in part by the NSF (USA) under grant DMS-1115118, the DFG (Germany) under grant KA 3215/2-1, and the Darmstadt Graduate Schools of Excellence Computational Engineering and Energy Science and
Engineering.
Date
2016-05-26
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Society for Industrial and Applied Mathematics
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Keywords
Finite element method, Anisotropic mesh, Stability condition, Parabolic equation, Explicit one-step method
Citation
Huang, W., Kamenski, L., & Lang, J. (2016). Stability of Explicit One-Step Methods for P1-Finite Element Approximation of Linear Diffusion Equations on Anisotropic Meshes. SIAM Journal on Numerical Analysis, 54(3), 1612-1634. doi:10.1137/130949531