A Stable Cut Finite Element Method for Partial Differential Equations on\n Surfaces: The Helmholtz-Beltrami Operator

2018/10/09 by Erik Burman, Peter Hansbo, Burman, Erik +5 · 1 citation
Engineering · Computer Science · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1810.04217

Abstract

We consider solving the surface Helmholtz equation on a smooth two\ndimensional surface embedded into a three dimensional space meshed with\ntetrahedra. The mesh does not respect the surface and thus the surface cuts\nthrough the elements. We consider a Galerkin method based on using the\nrestrictions of continuous piecewise linears defined on the tetrahedra to the\nsurface as trial and test functions.Using a stabilized method combining\nGalerkin least squares stabilization and a penalty on the gradient jumps we\nobtain stability of the discrete formulation under the condition h k < C,\nwhere h denotes the mesh size, k the wave number and C a constant\ndepending mainly on the surface curvature \κ, but not on the surface/mesh\nintersection. Optimal error estimates in the H¹ and L²-norms follow.\n

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