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

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_xi)_xj + ∑_{i = 1}ⁿ bⁱ u_xi + 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 ⊄ H₀¹ (Ω)). We assume that ∥ v ∥_{L2 (∂ Ω)} ≤ Ch_μ^m ∥ v ∥_{H1 (Ω)}, for all v ∈ S_μ, and there exists u_I ∈ S_μ such that ∥ u - u_I ∥_{H1 (Ω)} ≤ Ch_μ^j ∥ u ∥_{Hj + 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_xi)_xj + ∑_{i = 1}ⁿ bⁱ u_xi + cu = f in Ω, u = g on ∂ Ω. Finally, we outline the construction of a sequence of GFEM spaces S_μ ⊂ over(S, ̃)_μ, μ = 1, 2, ..., that satisfies our assumptions.