Anisotropic graded meshes and quasi-optimal rates of convergence for the FEM on polyhedral domains in 3D

We consider the model problem -Δu + V u = f ∈ Ω with suitable boundary conditions on ∂Ω where Ω is a bounded polyhedral domain in ℝ^d, d = 2; 3, and V is a possibly singular potential. We study efficient finite element discretizations of our problem following Numer. Funct. Anal. Optim., 28(7-8):775-824, 2007, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55(103):157-178, 2012, and a few other more recent papers. Under some additonal mild assumptions, we show how to construct sequences of meshes such that the sequence of Galerkin approximations u _κ ∈ S_κ achieve h^m-quasi- optimal rates of convergence in the sense that ∥u - u_κ ∥H¹(Ω) ≤ C dim(S_κ) ^m/d∥f∥H^m-1(Ω). Our meshes defining the Finite Element spaces Sk are suitably graded towards the singularities. They are topologically equivalent to the meshes obtained by uniform refinement and hence their construction is easy to implement. We explain in detail the mesh refinement for four typical problems: for mixed boundary value/transmission problems for which V = 0, for Schrödinger type operators with an inverse square potential V on a polygonal domain in 2D, for Schrödinger type operators with a periodic inverse square potential V in 3D, and for the Poisson problem on a three dimensional domain. These problems are listed in the increasing order of complexity. The transmission problems considered here include the cases of multiple junction points.