A Fourth-order KFBI Method For Elliptic PDEs And Its Applications

Fourth-order kernel-free boundary integral(KFBI)method serves as a fast,stable and high-accurate numerical method for solving elliptic partial differential equations in irregular domains.It is a combination of boundary integral method and Cartesian grid method,but different from the traditional boundary integral method which depends on Green's function.Meanwhile,it facilitates the Cartesian grid method when describing those irregular domains.This dissertation will present detailed contents for the KFBI method from the aspects of research background,theoretical basis,core algorithm,related applications and its extension.The KFBI method is an improvement and development for the traditional boundary integral method.According to the potential theory,the original elliptic boundary value problem(BVP)can be reformulated into the corresponding boundary integral equation,whose discrete form can be solved by the Krylov subspace method.This work employs two kinds of subspace methods,the Richardson iteration and the generalized minimum residual(GMRES)method.To evaluate the volume or boundary integrals in the equation,traditional boundary integral method needs the exact expression of the involved kernel,whereas the KFBI method resorts to solving their equivalent interface problems.For this end,this dissertation will introduce two classes of Cartesian grid methods,i.e.,the fourth-order finite difference method and the second-order tailored finite point method.Different methods for interface problems in different dimensions all contain four essential steps,namely discretizing the control equation with some compact schemes,making corrections for the scheme at irregular grid nodes near the interface,solving the linear system with fast solvers and extracting the one-sided boundary data by some space interpolation techniques.In this work,the finite difference method adopts the nine-point or 27 point compact difference scheme for two or three dimensional cases respectively.The following correction process only changes the right hand side of the scheme while remains the left hand side unchanged,the resulting linear system is then solved with a fast Fourier transform based fast elliptic solver.At last,boundary datas are obtained by quartic polynomial interpolation.For tailored finite point method,the corresponding tailored finite point sheme,tailored finite point correction as well as tailored finite point interpolation can be similarly constructed based on the modified Bessel function expansion or exponential function combination.The linear system attained in this way is iteratively solved by some general treatments such as the preconditioned conjugate gradient method.The remaining part of this dissertation will focus on the fourth-order finite difference integrated KFBI method and its application,including the direct application on solving the second-order elliptic BVPs and the fourth-order biharmonic BVPs in two space dimensions,the second-order elliptic BVPs in three space dimensions,as well as the acoustic multiple scattering problems of the Helmholtz equation in complex space,and also the indirect application on Stokes and Navier-Stokes equations for incompressible fluid,where the KFBI method,serving as the spatial discretization method,is combined with the fourth-order composite backward differentiation formula or the third-order semi-implicit Runge-Kutta method for time integtation.Especially,this dissertation also gives the KFBI method for solving the singular perturbation problem,where the equivalent interface problem for integral evaluation is dealt with the second-order tailored finite point method.The proposed method is utilized to solve the singular perturbated reaction-diffusion equation with the second-order semi-implicit RungeKutta method used for time discretization.