Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections

Abstract

Consider Dirichlet’s problem in a plane domain Ω \Omega with smooth boundary ∂ Ω \partial \Omega . For the purpose of its approximate solution, an approximating domain Ω h , 0 > h ≦ 1 {\Omega _h},0 > h \leqq 1 , with polygonal boundary ∂ Ω h \partial {\Omega _h} is introduced where the segments of ∂ Ω h \partial {\Omega _h} have length at most h h . A projection method introduced by Nitsche [6] is then applied on Ω h {\Omega _h} to give an approximate solution in a finite-dimensional subspace of functions S h {S_h} , for instance a space of splines defined on a triangulation of Ω h {\Omega _h} . The boundary terms in the bilinear form associated with Nitsche’s method are modified to correct for the perturbation of the boundary.