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 ℝd 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.
Funder
Engineering and Physical Sciences Research Council
Stiftelsen för Strategisk Forskning
Vetenskapsrådet
Swedish strategic research programme eSSENCE
Subject
Applied Mathematics,Modelling and Simulation,Numerical Analysis,Analysis,Computational Mathematics
Cited by
33 articles.
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