Boundary penalty techniques

We examine techniques for enforcing Dirichlet boundary data in finite element solution of boundary-value problems. The common programming strategy of adding a large number to certain diagonal entries and scaling the right-side is compared with penalty methods. By selectively underintegrating the continuous penalty formulation and selecting the penalty parameter ε^?1 optimally as a function of mesh size h we deduce the ‘large number’ strategy and hence show that the optimal rate is preserved. Numerical experiments to study the rates for continuous penalty, underintegrated penalty and ‘large number’ strategies corroborate the argument.