Approximate Dirichlet Boundary Conditions in the Generalized Finite Element Method

We propose a method for treating the Dirichlet boundary conditions in the framework of the Generalized Finite Element Method (GFEM). We use approximate Dirichlet boundary conditions as in [12] and polynomial approximations of the boundary. Our sequence of GFEM-spaces considered, Sμ, μ = 1, 2, . . . is such that Sμ 6 ⊂ H1 0 (Ω), and hence it does not conform to one of the basic FEM conditions. Let hμ be the typical size of the elements defining Sμ and let u ∈ Hm+1(Ω) be the solution of the Dirichlet problem −∆u = f in Ω, u = 0 on ∂Ω, on a smooth, bounded domain Ω. Assume that ‖vμ‖H1/2(∂Ω) ≤ Ch m μ ‖vμ‖H1(Ω) for all vμ ∈ Sμ and |u − uI |H1(Ω) ≤ Chμ ‖u‖Hm+1(Ω), u ∈ Hm+1(Ω) ∩ H1 0 (Ω), for a suitable uI ∈ Sμ. Then we prove that we obtain quasi-optimal rates of convergence for the sequence uμ ∈ Sμ of GFEM approximations of u, that is, ‖u− uμ‖H1(Ω) ≤ Chμ ‖f‖Hm−1(Ω). Next, we indicate an effective technique for constructing sequences of GFEM-spaces satisfying our conditions using polynomial approximations of the boundary. Finally, we extend our results to the inhomogeneous Dirichlet boundary value problem −∆u = f in Ω, u = g on ∂Ω.