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

✍ Scribed by Ivo Babuška; Victor Nistor; Nicolae Tarfulea

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 182 KB
• Volume: 218
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2007.04.041

✦ Synopsis

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

where is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, {S } 1 , which are nonconforming (that is S / ⊂ H 1 0 ( )). We assume that v L 2 (j ) Ch m v H 1 ( ) , for all v ∈ S , and there exists u I ∈ S such that uu I H 1 ( ) Ch j u H j +1 ( ) , 0 j m, where u ∈ H m+1 ( ) is the exact solution, m is the expected order of approximation, and h is the typical size of the elements defining S . Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence u ∈ S of u. Next, we extend our analysis to the inhomogeneous boundary value problem -n i,j =1 (a ij u x i ) x j + n i=1 b i u x i + cu = f in , u = g on j . Finally, we outline the construction of a sequence of GFEM spaces S ⊂ S , = 1, 2, . . . , that satisfies our assumptions.