  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.
