Fast Algorithm for the Fourth-Order Elliptic Problem Based on Orthogonal Matrix Decomposition

A fast algorithm for solving the first biharmonic boundary problem on a rectangular domain is presented. It is based on splitting the fourth-order problem into a coupled system of two-order problems, whose finite-di↵erence approximations are solved iteratively with linear algebra routines. There, a crucial role plays the orthogonal eigenvalue decomposition of the iteration matrix, which leads to a reduction of the operational count for one iteration to the asymptotically optimal value. This approach is extensively tested from the point of view of the total computational time, number of iterations and the solution error. It is shown, that these values cope with the theoretical assumptions and scale convincingly up to a problem with 100 million grid points.