Isoparametric finite element analysis of a generalized Robin boundary value problem on curved domains

Abstract

We study the discretization of an elliptic partial differential equation, posed on a two- or three-dimensional domain with smooth boundary, endowed with a generalized Robin boundary condition which involves the Laplace-Beltrami operator on the boundary surface. The boundary is approximated with piecewise polynomial faces and we use isoparametric finite elements of arbitrary order for the discretization. We derive optimal-order error bounds for this non-conforming finite element method in both $L^2$- and $H^1$-norm. Numerical examples illustrate the theoretical results.