Babus̆ka’s Penalty Method for Inhomogeneous Dirichlet Problem: Error Estimates and Multigrid Algorithms

where gh is an approximation of g and V 0 h ⊂ H 1 0 (Ω) is a finite element subspace. In practice, gh is considered to be a nodal interpolation of g in Vh|∂Ω, where Vh ⊂ H(Ω) is a finite element subspace. However it is shown in [4] that the error estimates for (3) are better when gh is chosen to be the L2-projection of g onto Vh|∂Ω. In particular when gh is chosen to be a nodal interpolation, the L2 error estimate requires g to be a piecewise H function on ∂Ω, see [4, Theorem 7.1]. Perhaps this estimate may not be improved. On the other-hand a mild disadvantage associated with the choice that gh is L2-projection, we need to solve a system before we solve for the solution uh. An alternative method due to Babus̆ka [1] is to pose the boundary condition weakly by penalty. Namely, find uh ∈ Vh such that ah(uh, v) = Lh(v) ∀v ∈ Vh, (4)