A Highly Accurate Numerical Solution of a Biharmonic Equation

The coefficients for a nine-point high-order accuracy discretization scheme for a biharmonic equation ∇⁴ u = f(x, y) (∇² is the two-dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂² u/∂n² or (2) u and ∂u/∂n (where ∂/∂n is the normal to the boundary derivative) are specified at the boundary. For both considered cases, the truncation error for the suggested scheme is of the sixth-order O(h⁶) on a square mesh (h_x = h_y = h) and of the fourth-order O(h⁴_x, h²_xh²_y, h⁴_y) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The advantage of the suggested scheme is demonstrated for solving problems of the deflection of rectangular plates for cases of different boundary conditions: (1) a simply supported plate and (2) a plate with built-in edges. In order to demonstrate the high-order accuracy of the method, the numerical results are compared with exact solutions.