Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated

Inspired by the methods developed in J. H. Bramble, T. Dupont, and V. Thomée (“Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections,” Math. Comput. , vol. 26, no. 120, pp. 869–879, 1972), we introduce a new technique that yields a symmetric formulation and has similar performance. The new method is based on a Robin-type problem on an approximate polygonal domain. Optimal error estimates in the energy norm are proved for piecewise quadratics and cubics. We provide numerical experiments that show our theoretical results are sharp.