Computing eigenmodes ofelliptic operators using radial basis functions

Radial basis function (RBF) approximations have been successfully used to solve boundary-value problems numerically. We show that RBFs can also be used to compute eigenmodes of elliptic operators. Particular attention is given to the Laplacian operator in two dimensions, including techniques to avoid degradation of the solution near the boundaries. For regions with corner singularities, special functions must be added to the basis to maintain good convergence. Numerical results compare favorably to basic finite-element methods.