Title: A family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes
Abstract: In this paper, we propose and analyze a family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes. The present work covers a previous one Zhou and Wu (2020) where only the case of scalar diffusion coefficients is considered. The novelty of this paper is that, by the introduction of some special parameters involving the diffusion tensor and the mesh geometry, each entry of the element stiffness matrix can be split as three parts that are easily computed and analyzed. Thanks to this representation, we manage to obtain a sufficient condition that ensures the existence, uniqueness and coercivity result of the finite volume element solution on triangular mesh. Moreover, this sufficient condition has a simple, analytic and computable expression that can be easily judged on any triangular meshes with arbitrary full diffusion tensors. In particular, when the diffusion tensor is an identity matrix, this condition reduces to a geometric one and only relies on the interior angles of each triangular element, which is consistent with the result in (Zhou and Wu, 2020). Moreover, based on this sufficient condition, we obtain a minimum angle condition for the case of general diffusion tensors, which is better than some existing ones. By the coercivity result, the optimal H1 error estimate is also obtained. Finally, the theoretical results are verified by some numerical experiments.
Publication Year: 2021
Publication Date: 2021-09-03
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 3
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