The Bulk Trace FEM for the simultaneous solution of structural membranes on all level sets over a bulk domain

Abstract A finite element method for the simultaneous solution of boundary value problems (BVPs) posed on all level sets over some bulk domain is proposed. A level‐set function is used to define ‐dimensional curved manifolds which are bounded by the d ‐dimensional bulk domain. The BVPs considered herein model structural ropes and membranes in a large displacement context, resulting in a geometrically non‐linear mechanical model. For the numerical analysis, the bulk domain is discretized using d ‐dimensional finite elements which do not conform to the level sets implying the domains of interest. However, they conform to the boundary of the bulk domain. The proposed variant of the FEM is a hybrid between classical and non‐conforming FEMs and may be labeled Bulk Trace FEM. Numerical studies confirm optimal higher‐order convergence rates for the proposed method when simultaneously approximating BVPs on all level sets.