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

2008 ◽  
Vol 218 (1) ◽  
pp. 175-183 ◽  
Author(s):  
Ivo Babuška ◽  
Victor Nistor ◽  
Nicolae Tarfulea
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Elliptic Operators ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Generalized Finite Element Method ◽  
Second Order Elliptic Operators ◽  
Generalized Finite Element

APPROXIMATE AND LOW REGULARITY DIRICHLET BOUNDARY CONDITIONS IN THE GENERALIZED FINITE ELEMENT METHOD

2007 ◽  
Vol 17 (12) ◽  
pp. 2115-2142 ◽  
Author(s):  
IVO BABUŠKA ◽  
VICTOR NISTOR ◽  
NICOLAE TARFULEA
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Boundary Data ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Generalized Finite Element Method ◽  
Low Regularity ◽  
Generalized Finite Element ◽  
Element Method

We propose a method for treating Dirichlet boundary conditions for the Laplacian in the framework of the Generalized Finite Element Method (GFEM). A particular interest is taken in boundary data with low regularity (possibly a distribution). Our method is based on using approximate Dirichlet boundary conditions and polynomial approximations of the boundary. The sequence of GFEM-spaces consists of nonzero boundary value functions, and hence it does not conform to one of the basic Finite Element Method (FEM) conditions. We obtain quasi-optimal rates of convergence for the sequence of GFEM approximations of the exact solution. We also extend our results to the inhomogeneous Dirichlet boundary value problem, including the case when the boundary data has low regularity (i.e. is a distribution). Finally, we indicate an effective technique for constructing sequences of GFEM-spaces satisfying our assumptions by using polynomial approximations of the boundary.

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