Singular Function Enhanced Mortar Finite Element 477

We assume the function f ∈ L(Ω). We also assume the function u0 has an extension H(Ω), which we denote also by u0. We let the domain Ω to be the L-shaped domain in 2 with vertices V1 = {0, 0}, V2 = {1, 0}, V3 = {1, 1}, V4 = {−1, 1}, V5 = {−1,−1}, and V6 = {0,−1}. It is well-known that the solution u∗ of (1.1) does not necessarily belong to H(Ω) due to the nonconvexity of the domain Ω at the corner V1, and therefore, standard finite element discretizations do not give second order accurate schemes. Theoretical and numerical work on corner singularity are very well-known and several different approaches were proposed [4, 2, 5, 6, 7, 8, 9, 10]; see the references therein. The main goal of the paper is to design and analyze optimal accurate finite element discretizations based on mortar techniques [1, 11] and singular functions [8, 7]. The proposed methods are variation of the methods described in Chapter 8 of [10] where a smoothed cut-off singular function is added to the space of finite elements. There, a smoothed cut-off function is applied to make the singular function to satisfy the zero Dirichlet boundary condition. Here, instead, we use mortar finite element techniques on the boundary of ∂Ω to force, in a weak sense, the boundary condition. As a result, accurate and general schemes can be obtained for which they do not rely on costly numerical integrations and linear solvers.