Ambient residual penalty approximation of partial differential equations on embedded submanifolds

Abstract

In this paper, we present a novel approach to the approximate solution of elliptic partial differential equations on compact submanifolds of $\mathbb {R}^{d}$ , particularly compact surfaces and the surface equation ${\Delta }_{\mathbb {M}} u - \lambda u=f$ . In the course of this, we reconsider differential operators on such submanifolds to deduce suitable penalty based functionals. These functionals are based on the residual of the equation in an integral representation, extended by a penalty on the first-order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by numerical examples employing tensor product B-splines.