Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions

Abstract

We introduce a method for approximating essential boundary conditions—conditions of Dirichlet type—within the generalized finite element method (GFEM) framework. Our results apply to general elliptic boundary value problems of the form - ∑ i , j = 1 n ( a ij u x i ) x j + ∑ i = 1 n b i u x i + cu = f in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, { S μ } μ ⩾ 1 , which are nonconforming (that is S μ ⊄ H 0 1 ( Ω ) ). We assume that ∥ v ∥ L 2 ( ∂ Ω ) ⩽ Ch μ m ∥ v ∥ H 1 ( Ω ) , for all v ∈ S μ , and there exists u I ∈ S μ such that ∥ u - u I ∥ H 1 ( Ω ) ⩽ Ch μ j ∥ u ∥ H j + 1 ( Ω ) , 0 ⩽ j ⩽ m , where u ∈ H m + 1 ( Ω ) is the exact solution, m is the expected order of approximation, and h μ is the typical size of the elements defining S μ . Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence u μ ∈ S μ of u. Next, we extend our analysis to the inhomogeneous boundary value problem - ∑ i , j = 1 n ( a ij u x i ) x j + ∑ i = 1 n b i u x i + cu = f in Ω , u = g on ∂ Ω . Finally, we outline the construction of a sequence of GFEM spaces S μ ⊂ S ˜ μ , μ = 1 , 2 , … , that satisfies our assumptions.