We present a robust and effective method for the numerical solution of the biharmonic interface problem with discontinuities in both the solution and its derivatives. We use a mixed scheme, in which the biharmonic equation is decoupled to two Poisson equations. The proposed approach is based on the method of difference potentials combined with finite difference schemes on regular structured grid to solve this problem with high‐order accuracy on nonconforming domains. Representative numerical experiments confirm the accuracy and effectiveness of the proposed method and its ability to handle problems with coupled equations.
In this paper, we present a robust and accurate numerical algorithm for solving the nonlinear Poisson-Boltzmann equation, based on the difference potentials method (DPM). First, we remove singularity by a simple regular-ization technique and employ Newton’s method for linearizing the equation. Then we use the difference potentials method combined with a second-order finite difference scheme and curvilinear approach to solve the problem in the regions with a smooth general-shaped interface. Unlike many other methods, DPM does not need to treat the nonhomogeneous interface conditions resulting from the regularization and can handle discontinuity in the interface without loss of accuracy. In DPM the grid does not need to conform to the boundary or interfaces. This method approximates the resulting equation on a regular structured grid, which entails a low computational complexity, and does not face the challenge of reducing the order of accuracy near the nonconforming interfaces. Several numerical experiments are presented for illustrating the efficiency and accuracy of the developed numerical method.
Mathematics Subject Classification (2000) 41A58 · 82B24 · 65N06 · 35J66
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