Abstract
The boundary collocation method is applied to the computation of eigenvalues and eigenfunctions of the Laplace operator on planar simply connected regions with smooth boundaries. For convex regions we seek to approximate the eigenfunctions by a linear combination of basis functions that contain Bessel functions of the first kind. Our method differs from related schemes proposed previously in that we distribute the collocation points differently, and we use a different iterative scheme for computing eigenvalues and eigenfunctions. This makes our method both faster and more accurate. For nonconvex regions rapid convergence generally can be achieved only if the eigenfunctions are approximated by functions with singular points in the finite plane. A boundary collocation method with such basis functions is also described.
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Reichel, L. (1987). Boundary collocation in Fejér points for computing eigenvalues and eigenfunctions of the Laplacian. In: Saff, E.B. (eds) Approximation Theory, Tampa. Lecture Notes in Mathematics, vol 1287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078903
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DOI: https://doi.org/10.1007/BFb0078903
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