A Fast Boundary Integral Method Solving The Dirichlet Problem Of Poisson Equation

The Poisson equation is an elliptic partial differential equation with broad applications in mechanical engineering and theoretical physics.It ordinarily arises in various application fields,such as fluid dynamics,acoustics,heat transfer,electromagnetics,electrostatics,mechanical engineering,etc.Hence,there have been extensive research activities in the past decades,including theoretical and computational,on the Poisson equation.In this paper,we develop a fast boundary integral equation method to solve the Dirichlet boundary value problem of the Poisson equation.To this end,we propose a high-accuracy algorithm for calculating Newton's potential at first.Then,a fast Fourier-Galerkin method,solving the boundary integral equation deduced from the Poisson equation,is presented.The convergence order and computation cost of the fast Fourier Galerkin method are analyzed and demonstrated by numerical experiments.In Chapter 2,we introduce two kinds of multidimensional integration methods.One method is the composite seven-point Gauss-Legendre(CGL)method,widely used in the Gauss method.Another method is the Tanh-Sinh(TS)quadrature formula in the double exponential integration method(DE),which is specialized in solving the singular integrals of the endpoints.We derive the quadrature formula and the error estimation formula of the CGL method in the multidimensional finite domain.We establish a quadrature formula for computing the 2D integrals with edge singularity.This formula is a hybrid of the CGL method and the TS quadrature.We prove that the quadrature formula's accuracy is O(n-14),which is competitive with the CGL method for the smooth integrand.Meanwhile,the total number of additions and multiplications required is O(n log2 n),where n represents the number of sub-integral regions divided by the CGL method.Numerical examples are presented to confirm the theoretical results.In Chapter 3,we present a fast Fourier-Galerkin(FFG)method solving the two-dimensional Poisson equation.Compared with the existing FFG methods,the right-hand side of the boundary integral equation will involve evaluating the Fourier coefficients of the Newton's potential,a region integral with a singular kernel.This will take large computational costs.To handle this issue,we present an algorithm to calculate the Fourier coefficients of the Newton's potential with high accuracy.We prove that the proposed FFG method enjoys the same optimal convergence order O(n-t).Meanwhile,the total number of multiplications used to generate the linear system for the proposed method is quasi-linear O(n log3 n),where n represents the maximum number of order of the Fourier basis function used,the t represents the order of regularity of the exact solution.A numerical example is given to verify the approximation accuracy and computational efficiency of the proposed method.