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

TL;DR

This paper develops an isoparametric finite element method for elliptic PDEs with generalized Robin boundary conditions involving surface Laplacians, providing optimal error bounds and numerical validation.

Contribution

It introduces a novel finite element approach for elliptic problems with complex boundary conditions on curved domains, including error analysis and numerical demonstrations.

Findings

Optimal error bounds in L2 and H1 norms

Effective discretization of boundary Laplace-Beltrami operator

Numerical results confirm theoretical accuracy

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.