A Priori Analysis of an Anisotropic Finite Element Method for Elliptic Equations in Polyhedral Domains

Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces H-sigma with sigma is an element of [0, 1). We propose anisotropic, finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with fewer geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires the pure Dirichlet boundary condition. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations with the price to have "smoother" interior data. Numerical tests validate the theoretical analysis.