A Stable Cut Finite Element Method for Partial Differential Equations on Surfaces: The Helmholtz-Beltrami Operator
Autor: | Burman, Erik, Hansbo, Peter, Larson, Mats G., Massing, Andre |
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Rok vydání: | 2018 |
Předmět: | |
Druh dokumentu: | Working Paper |
Popis: | We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition $h k < C$, where $h$ denotes the mesh size, $k$ the wave number and $C$ a constant depending mainly on the surface curvature $\kappa$, but not on the surface/mesh intersection. Optimal error estimates in the $H^1$ and $L^2$-norms follow. Comment: 27 pages, 10 figures |
Databáze: | arXiv |
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