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An h-narrow band finite element method for elliptic equations on implicit surfaces

by Deckelnick, K., Dziuk, G., Elliott, C.M., Heine, C.-J..

Series: 2007-50, Preprints

65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65N12 Stability and convergence of numerical methods
35J70 Degenerate elliptic equations

In this article we define a finite element method for elliptic partial differential equations on curves or surfaces, which are given implicitly by some level set function. The method is specially designed for complicated surfaces. The key idea is to solve the PDE on a narrow band around the surface. The width of the band is proportional to the grid size. We use piecewise linear finite element spaces which are unfitted to the narrow band, so that elements are cut off. The implementation nevertheless is easy. We prove error estimates of optimal order for a Poisson equation on a surface and provide numerical tests and examples.

elliptic equations, implicit surfaces, level sets, unfitted mesh finite element method, error estimates