Unfitted finite element methods using bulk meshes for surface partial differential equations (Q2927849)

The authors consider the finite element approximation of surface partial differential equations. They define new unfitted finite element methods for numerically approximating the exact solution using bulk finite elements. The key idea is that the \(n\)-dimensional hypersurface, \(\Gamma \subset \mathbb{R}^{n+1}\), is embedded in a polyhedral domain in \(\mathbb R^{n+1}\), consisting of a union, \(\mathcal{T}_h\), of \((n+1)\)-simplices. The unifying feature of the methodological approach is that the finite element approximating space is based on continuous piecewise linear finite element functions on the bulk triangulation \(\mathcal{T}_h\) which is independent of \(\Gamma\). The first method is a sharp interface method (\textit{SIF}) which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, \(\Gamma_{h}\), of \(\Gamma\). The full gradient is used rather than the projected tangential gradient and it is this which distinguishes \textit{SIF} from some previous methods. The second method is a narrow band method (\textit{NBM}) in which the region of integration is a narrow band of width \(O(h)\). \textit{NBM} is similar to the method of \textit{K. Deckelnick} et al. [IMA J. Numer. Anal. 30, No. 2, 351--376 (2010; Zbl 1191.65151)], but again, the full gradient is used in the discrete weak formulation. An a priori error analysis is presented and it shows that the methods are of optimal order in the surface \(L^{2}\) and \(H^{1}\) norms. The third method combines bulk finite elements, discrete sharp interfaces, and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. It is shown that the method is conservative so that it preserves mass in the case of an advection-diffusion conservation law. Some numerical results are presented to illustrate the rates of convergence.