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

Abstract:

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