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


Author: Hengguang Li
Journal: Math. Comp.
MSC (2010): Primary 65N15, 65N30, 65N50; Secondary 35J15, 35J75
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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Additional Information

Hengguang Li
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: li@wayne.edu

DOI: https://doi.org/10.1090/mcom/3290
Received by editor(s): July 12, 2016
Received by editor(s) in revised form: March 1, 2017
Published electronically: October 31, 2017
Additional Notes: The author was supported in part by the NSF Grant DMS-1418853, by the Natural Science Foundation of China Grant 11628104, and by the Wayne State University Grants Plus Program
Article copyright: © Copyright 2017 American Mathematical Society