A Novel Finite-Difference Approximation to the Biharmonic Operator

The two-dimensional biharmonic operator is approximated by a finite-difference operator over a square (h x A) net, which connects each node with 16 neighbouring nodes in such a manner that the resulting matrix has "Young's Property A," for simple boundary conditions. It is shown that the local truncation error, the convergence rate of S.O.R. for solving the finite-difference (or "net") equations, and the truncation error of the solution of the net equations are each of order O(A) as h -> 0. Comparisons are made with the conventional 13-node approximation to the biharmonic operator: in particular, numerical experiments indicate that the convergence rate of S.O.R. for solving net equations based on the conventional 13-node operator is considerably less than that for the novel 17-node operator.