A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains

Abstract

For elliptic boundary value problems (BVPs) involving irregular domains and Robin boundary condition, no numerical method is known to deliver a fourth order convergence and O(Nlog⁡N) efficiency, where N stands for the total degree-of-freedom of the system. Based on the matched interface and boundary (MIB) and fast Fourier transform (FFT) schemes, a new finite difference method is introduced for such problems, which involves two main components. First, a ray-casting MIB scheme is proposed to handle different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mix combinations. By enclosing the concerned irregular domain by a large enough cubic domain, the ray-casting MIB scheme generates necessary fictitious values outside the irregular domain by imposing boundary conditions along the normal direction of the boundary, so that a high order central difference discretization of the Laplacian can be formed. Second, an augmented MIB formulation is built, in which Cartesian derivative jumps are reconstructed on the boundary as auxiliary variables. By treating such variables as unknowns, the discrete Laplacian can be efficiently inverted by the FFT algorithm, in the Schur complement solution of the augmented system. The accuracy and efficiency of the proposed augmented MIB method are numerically examined by considering various elliptic BVPs in two and three dimensions. Numerical results indicate that the new algorithm not only achieves a fourth order of accuracy in treating irregular domains and complex boundary conditions, but also maintains the FFT efficiency.