Unfitted Finite Element Methods Using Bulk Meshes for Surface Partial Differential Equations

In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ ⊂ Rn+1, is embedded in a polyhedral domain in Rn+1 consisting of a union, Th, 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 Th which is independent of Γ. Our first method is a sharp interface method (SIF ) which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γh, of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes SIF from the method of [M. A. Olshanskii, A. Reusken, and J. Grande, SIAM J. Numer. Anal., 47 (2009), pp. 3339–3358]. The second method is a narrow band method (NBM ) in which the region of integration is a narrow band of width O(h). NBM is similar to the method of [K. Deckelnick et al., IMA J. Numer. Anal., 30 (2010), pp. 351–376] but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L2 and H1 norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our 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. We show that our method is conservative so that it preserves mass in the case of an advection-diffusion conservation law. Numerical results are given which illustrate the rates of convergence.