A new anisotropic finite element method on polyhedral domains: interpolation error analysis

TL;DR

This paper introduces a new anisotropic finite element method with specialized mesh refinement algorithms for solving the Poisson equation on polyhedral domains, enhancing convergence for singular solutions.

Contribution

It presents a simple, explicit anisotropic mesh refinement algorithm with relaxed geometric conditions and develops interpolation error estimates for better approximation of singular solutions.

Findings

Improved convergence for singular solutions on polyhedral domains.

Validated the effectiveness of the proposed mesh refinement through numerical tests.

Provided error estimates in weighted spaces for anisotropic meshes.

Abstract

Consider the Poisson equation with the Dirichlet boundary condition on a three-dimensional polyhedral domain. For singular solutions from the non-smoothness of the domain boundary, we propose new anisotropic tetrahedral 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. These estimates can be used to design optimal finite element methods approximating singular solutions. We report numerical test results to validate the method.