Least squares collocation solution of elliptic problems in general regions
Mathematics and Computers in Simulation - Special issue: Applied and computational mathematics - selected papers of the fifth PanAmerican workshop - June 21-25, 2004, Tegucigalpa, Honduras
Eigenfrequencies of fractal drums
Journal of Computational and Applied Mathematics
Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids
Computer-Aided Design
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Optimization of spectral functions of Dirichlet-Laplacian eigenvalues
Journal of Computational Physics
Benchmark results for testing adaptive finite element eigenvalue procedures
Applied Numerical Mathematics
Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues
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Meshfree Particle Methods in the Framework of Boundary Element Methods for the Helmholtz Equation
Journal of Scientific Computing
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Fox, Henrici, and Moler made famous a "method of particular solutions" for computing eigenvalues and eigenmodes of the Laplacian in planar regions such as polygons. We explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties. The crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant. Similar methods may be used to improve other "mesh-free" algorithms for a variety of computational problems.