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=1n(aijuxi)xj+∑i=1nbiuxi+cu=f in ΩΩ, u=0u=0 on ∂Ω∂Ω, where ΩΩ is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, {Sμ}μ⩾1{Sμ}μ⩾1, which are nonconforming (that is Sμ⊄H01(Ω)). We assume that ∥v∥L2(∂Ω)⩽Chμm∥v∥H1(Ω), for all v∈Sμv∈Sμ, and there exists uI∈SμuI∈Sμ such that ∥u-uI∥H1(Ω)⩽Chμj∥u∥Hj+1(Ω), 0⩽j⩽m0⩽j⩽m, where u∈Hm+1(Ω)u∈Hm+1(Ω) is the exact solution, m   is the expected order of approximation, and hμhμ is the typical size of the elements defining SμSμ. Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence uμ∈Sμuμ∈Sμ of u  . Next, we extend our analysis to the inhomogeneous boundary value problem -∑i,j=1n(aijuxi)xj+∑i=1nbiuxi+cu=f in ΩΩ, u=gu=g on ∂Ω∂Ω. Finally, we outline the construction of a sequence of GFEM spaces Sμ⊂S˜μ, μ=1,2,…,μ=1,2,…, that satisfies our assumptions.