An Augmented Method for 4th Order PDEs with Discontinuous Coefficients

Abstract

A fast finite difference method is developed for solving 4th order partial differential equations with discontinuous coefficients across an arbitrary interface. The method is based on an augmented approach by introducing an intermediate (augmented variable) boundary condition $\varDelta u|_{\partial \varOmega }$ along the boundary so that the problem can be treated as two separated Poisson equations with jumps in the source terms along the interface. Thus a fast Poisson solver can be utilized, which makes the proposed method fast. The augmented variable should be chosen such that the original boundary condition $\frac{\partial u}{\partial n}|_{\partial \varOmega }$ is satisfied. In discretization, the augmented variable is solved first using the Schur complement associated with the method. The proposed method is probably the first finite difference method for such an interface problem although the paper is partially motivated by the immersed finite element method (Lin et al in J Comput Appl Math 235(13):3953–3964, 2011) for such a problem. Numerical experiments against analytic solutions show that the computed solution using the proposed method has second order accuracy (convergence) in the maximum norm. Numerical results and analysis are also presented for various jump ratios and arbitrary interfaces.