Research Article

Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions

¹ Department of Mathematics, University College London, London, WC1E 6BT, UK
² Department of Mechanical Engineering, Jönköping University, 55111 Jönköping, Sweden
³ Department of Mathematics and Mathematical Statistics, Umeå University, 90187 Umeå, Sweden

^* Corresponding author: andre.massing@umu.se

Received: 20 September 2017
Accepted: 25 May 2018

Abstract

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in ℝ^d of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in ℝ³.