Article,
Finite element approximation of the Dirichlet problem using the boundary penalty method
Numerische Mathematik, (1986)
DOI: 10.1007/BF01389536
Abstract
This paper considers a finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a region Ω ⊂ ℝn (n=2 or 3) by the boundary penalty method. If the finite element space defined over Dh, a union of elements, has approximation power h K in the L 2 norm, then
(i)
for Ω≡D h convex polyhedral, we show that choosing the penalty parameter ε≡h λ with λ≧K yields optimal H 1 and L 2 error bounds if u∈H K+1(Ω);
(ii)
for ϖΩ being smooth, an unfitted mesh(Ω⊆Dh)
and assuming u∈H K+2(Ω) we improve on the error bounds given by Babuska 1. As (ii) is not practical we analyse finally a fully practical piecewise linear approximation involving domain perturbation and numerical integration. We show that the choice λ=2 yields an optimal H 1 and interior L 2 rate of convergence for the error. A numerical example is presented confirming this analysis.
