Finite element approximation of the Dirichlet problem using the boundary penalty method

Summary

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 overD ^h, a union of elements, has approximation powerh ^K in theL ² norm, then

• (i) for Ω≡D ^h convex polyhedral, we show that choosing the penalty parameter ε≡h ^λ with λ≧K yields optimalH ¹ andL ² error bounds ifu∈H ^K+1(Ω);

• (ii) for ϖΩ being smooth, an unfitted mesh$(\Omega \subseteq D^h )$ and assumingu∈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 optimalH ¹ and interiorL ² rate of convergence for the error. A numerical example is presented confirming this analysis.