An anisotropic finite element method on polyhedral domains: Interpolation error analysis

Li, Hengguang

doi:10.1090/mcom/3290

An anisotropic finite element method on polyhedral domains: Interpolation error analysis
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by Hengguang Li;

Math. Comp. 87 (2018), 1567-1600

DOI: https://doi.org/10.1090/mcom/3290

Published electronically: October 31, 2017

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Abstract:

On a polyhedral domain $\Omega \subset \mathbb R^3$, consider the Poisson equation with the Dirichlet boundary condition. For singular solutions from the non-smoothness of the domain boundary, we propose new anisotropic mesh refinement algorithms to improve the convergence of finite element approximation. The proposed algorithm is simple, explicit, and requires less geometric conditions on the mesh and on the domain. Then, we develop interpolation error estimates in suitable weighted spaces for the anisotropic mesh, especially for the tetrahedra violating the maximum angle condition. These estimates can be used to design optimal finite element methods approximating singular solutions. We report numerical test results to validate the method.

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