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

摘要

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 -Sigma(n)(i,j=1) (a(ij) u(xi))x(j) + Sigma(n)(i=1) b(i)u(xi) + cu = f in Omega, u=0 on partial derivative Omega, where Omega is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, {S-mu}(mu >= 1), which are nonconforming (that is S-mu not subset of H-0(1)(Omega)). We assume that parallel to nu parallel to(L2(partial derivative Omega)) <= Ch(mu)(m)parallel to nu parallel to(H1(Omega)), for all v is an element of S-mu, and there exists u(1) is an element of S-mu, such that parallel to u-u(1)parallel to(H1(Omega)) <= Ch(mu)(j)parallel to u parallel to(Hj+1(Omega)), 0 <= j <= m, where u is an element of Hm+1(Omega) is the exact solution, m is the expected order of approximation, and hp is the typical size of the elements defining S, Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence up E S, of u. Next, we extend our analysis to the inhomogeneous boundary value problem -Sigma(n)(i,j=1)(a(ij)u(xi))(xj) + Sigma(n)(i=1) b(i)u(xi) + cu = f in Omega, u =g on partial derivative Omega. Finally, we outline the construction of a sequence of GFEM spaces S-mu subset of (S) over tilde (mu), mu = 1, 2,..., that satisfies out assumptions. (C) 2007 Published by Elsevier B.V.