A Level-Set Method for Computing the Eigenvalues of Elliptic Operators Defined on Compact Hypersurfaces

Authors:
Jeremy Brandman
Affiliations:
Department of Mathematics, University of California Los Angeles, Los Angeles, USA 90095
Venue:
Journal of Scientific Computing
Year:
2008

Citing 9
Cited 3

Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations

Journal of Computational Physics
Weakly differentiable functions

Weakly differentiable functions
A simple level set method for solving Stefan problems

Journal of Computational Physics
Variational problems and partial differential equations on implicit surfaces

Journal of Computational Physics
Motion of curves constrained on surfaces using a level-set approach

Journal of Computational Physics
An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface

Journal of Scientific Computing
Fourth order partial differential equations on general geometries

Journal of Computational Physics
An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries

Journal of Scientific Computing
Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids

Computer-Aided Design

Variational piecewise constant level set methods for shape optimization of a two-density drum

Journal of Computational Physics
An Eulerian approach for computing the finite time Lyapunov exponent

Journal of Computational Physics
Solving eigenvalue problems on curved surfaces using the Closest Point Method

Journal of Computational Physics

Quantified Score

Hi-index	0.02

Visualization

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 驴 n can be found by solving an elliptic eigenvalue problem in a bounded domain 驴驴驴 n . 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).