Abstract
We demonstrate, through separation of variables and estimates from the semi-classical analysis of the Schrödinger operator, that the eigenvalues of an elliptic operator defined on a compact hypersurface in ℝn can be found by solving an elliptic eigenvalue problem in a bounded domain Ω⊂ℝn. The latter problem is solved using standard finite element methods on the Cartesian grid. We also discuss the application of these ideas to solving evolution equations on surfaces, including a new proof of a result due to Greer (J. Sci. Comput. 29(3):321–351, 2006).
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Brandman, J. A Level-Set Method for Computing the Eigenvalues of Elliptic Operators Defined on Compact Hypersurfaces. J Sci Comput 37, 282–315 (2008). https://doi.org/10.1007/s10915-008-9210-z
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DOI: https://doi.org/10.1007/s10915-008-9210-z