The Fast Algorithm For Boundary Integral Equation

The numerical solution to boundary integral equations is often constrained by the singularity of the integral operator and the coefficient matrix of the corresponding dis-crete linear system is often a dense matrix.These problems make the boundary integral equation method require more computing resources to solve large-scale problems.This paper considers the fast Fourier Galerkin method,this algorithm splits a singular boundary integral operator into the form of integral operator A+B,where operator A contains the most of the singularity of the original operator,and,the Fourier basis function is its eigen-functions such that the discrete matrix of operator A under the Fourier basis functions is a diagonal matrix.On the other hand,operator B has better smoothness than the original operator,and integral kernel is less singular than the original one,so that the discrete ma-trix of operator B under the Fourier basis function is only significantly non-zero in part of the region.By designing a proper truncation strategy,the dense matrix of the original operator under the Fourier basis can be compressed into a sparse matrix to achieve the pur-pose of increasing the calculation speed.According to the existing research,the number of non-zero elements in the coefficient matrix of the original operator can be reduced to the O?n log n?level,where n represents the maximum order of the Fourier basis function used.And the method can maintain the same optimal convergence order O(n^-t)as the traditional Fourier Galerkin method,where t represents the order of regularity of the true solution.In this paper,we improved the fast Fourier Galerkin method for solving a class of singular boundary integral equations,arising from the interior Dirichlet or Neumann boundary value problem for the Laplace equation,to reduce the number of non-zero el-ements of the discrete matrix to O?n?level.At the same time,the algorithm is applied to solve a matrix integral equation system which is a reformulation of the biharmonic equation,and the corresponding theoretical and numerical results are given.The specific research contents and conclusions are as follows:For the interior Dirichlet or Neumann boundary value problem of Laplace equation,we first use the potential theory to obtain the integral equation form of the original prob-lem,and then get the boundary integral equation.For the weakly singular and hypersin-gular boundary integral operator,we split the integral kernel to get two operators.In the process of compressing the matrix,we improved the truncation strategy of the discrete matrix,so that the number of non-zero elements in the matrix reached O?n?level,and give a theoretical proof.For the improved fast Fourier Galerkin method,we analyze the stability of the algorithm through Fredholm's theorem and perform a convergence analy-sis to prove that the approximate solution obtained by this proposed method preserves the same optimal convergence order O(n^-t)as the traditional Fourier Galerkin method.And the precondition method of the coefficient matrix is also given to control the condition number of the coefficient matrix.For the interior boundary value problem of the bihar-monic equation,we first give the boundary integral equation form of the original problem solution,and then convert it into a matrix integral equation system.Then we split the integral operator and design a suitable truncation strategy to truncate and compress the discrete matrix of the operator under the Fourier basis function,and give the theoretical stability and convergence analysis of this algorithm.At the end of each chapter,we give appropriate numerical examples to verify the accuracy and efficiency of our algorithm.