A Multigrid Method for Solving the Biharmonic Equation on Rectangular Domains
A simple and direct application of multigrid techniques for solving a standard finite difference approximation for the first boundary value problem of the biharmonic equation is described. Numerical results show that the convergence factors of the iterative multigrid method are h-independent and correspond to the good interior smoothing factors of the relaxation used. Moreover the convergence factors of our method are comparable to those of corresponding mu1tigrid iterations for the Poisson equation while its computational effort per iteration is higher by a factor of 2.3–3 (depending on the chosen variant). Furthermore numerical results for the full-multigrid method (FMG) are presented. As in the case of Poisson’s equation, FMG turns out to be an efficient approximate direct solver for the biharmonic equation with complexity O(N2) when solving the problem on a NxN grid, yielding solutions up to discretization errors.
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