The Numerical Solution of the Biharmonic Equation by Conformal Mapping

The solution to the biharmonic equation in a simply connected region Ω in the plane is computed in terms of the Goursat functions. The boundary conditions are conformally transplanted to the disk with a numerical conformal map. A linear system is obtained for the Taylor coefficients of the Goursat functions. The coefficient matrix of the linear system can be put in the form $I + K$, where K is the discretization of a compact operator. K can be thought of as the composition of a block Hankel matrix with a diagonal matrix. The compactness leads to clustering of eigenvalues, and the Hankel structure yields a matrix-vector multiplication cost of O(Nlog N). Thus, if the conjugate gradient method is applied to the system, then superlinear convergence will be obtained. Numerical results are given to illustrate the spectrum clustering and superlinear convergence.