Pseudospectral collocation methods for fourth-order differential equations

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multi-domain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C^1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C^3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multi-domain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a non-rectangular domain.