| This thesis work is aimed to develop a high-order accurate kernel-free bound-ary integral method for the Laplace equation in three-dimensional complex do-mains. The KFBI method is a Cartesian grid method. It solves the elliptic PDE in the framework of boundary integral equations. In the solution of the bound-ary integral equation, the KFBI method evaluates the involved boundary integral or volume integral by first solving an equivalent simple interface problem on a Cartesian grid and then interpolating the discrete data on the Cartesian grid to discretization points on the boundary or interface of the complex domain. The method is kernel-free and does not need to know the Green’s function or the kernel of the boundary/volume integral. It overcomes some typical limitations associated with the traditional boundary integral method. During the kernel-free evaluation of a boundary/volume integral, as the simple interface problem is discretized on a Cartesian grid, the discrete system at irregular grid nodes has large local truncation errors and has to be corrected for accurate solution. This work derives high-order estimates for the local truncation error at irregular grid nodes of the 27-point compact finite difference scheme for the Laplace equation and proposes the corre-sponding formula for correcting the right hand side of the discrete interface system. This method does not only modify the coefficient matrix of the discrete interface problem, the 27-point finite difference equations can be solved with a fast Fourier transform (FFT) based fast Poisson solver. For a simple test problem, provided the jumps of partial derivatives up to the fourth-order are known, a C computer pro-gram developed has successfully validated the correction formula. On some typical geometries, including sphere, ellipsoid, torus, numerical results demonstrate the correction to the discrete system yields fourth-order accurate numerical solution in the discrete maximum norm. |
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