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

SummaryThis 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 overDh, a union of elements, has approximation powerhK in theL2 norm, then(i)for Ω≡Dh convex polyhedral, we show that choosing the penalty parameter ε≡hλ with λ≧K yields optimalH1 andL2 error bounds ifu∈HK+1(Ω);(ii)for ϖΩ being smooth, an unfitted mesh $$(\Omega \subseteq D^h )$$ and assumingu∈HK+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 optimalH1 and interiorL2 rate of convergence for the error. A numerical example is presented confirming this analysis.