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Consider the Poisson equation with the Dirichlet boundary condition on athree-dimensional polyhedral domain. For singular solutions from thenon-smoothness of the domain boundary, we propose new anisotropic tetrahedralmesh refinement algorithms to improve the convergence of finite elementapproximation. The proposed algorithm is simple, explicit, and requires lessgeometric conditions on the mesh and on the domain. Then, we developinterpolation error estimates in suitable weighted spaces for the anisotropicmesh. These estimates can be used to design optimal finite element methodsapproximating singular solutions. We report numerical test results to validatethe method.

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