A Fast Direct Solver for the Biharmonic Problem in a Rectangular Grid

We present a fast direct solver methodology for the Dirichlet biharmonic problem in a rectangle. The solver is applicable in the case of the second order Stephenson scheme [34] as well as in the case of a new fourth order scheme, which is discussed in this paper. It is based on the capacitance matrix method ([10], [8]). The discrete biharmonic operator is decomposed into two components. The first is a diagonal operator in the eigenfunction basis of the Laplacian, to which the FFT algorithm is applied. The second is a low rank perturbation operator (given by the capacitance matrix), which is due to the deviation of the discrete operators from diagonal form. The Sherman-Morrison formula [18] is applied to obtain a fast solution of the resulting linear system of equations.