报告人：江颖（中山大学）

邀请人：王海永

报告时间：2021年11月11日（星期四）9:00-11:00

报告地点：腾讯会议ID:882 209 746

报告题目：A fast algorithm for solving the boundary integral equations derived from Poisson equation and Laplace equation in a domain with corners

报告摘要：This talk is about fast algorithms for solving boundary-integral equations based on sparse grids. Specifically, we develop schemes for approximating functions with singularity based on sparse grids. Then, we apply the schemes to solve the boundary integral equations derived from Poisson-equation, and the Dirichlet problem of Laplace equation in a domain with corners .

In the first part of the talk, we consider solving the boundary integral equation derived from the Poisson equation. Evaluating the Newton potential is crucial for efficiently solving the boundary integral equation of the Dirichlet boundary value problem of the Poisson equation. In the context of the Fourier–Garlerkin method for solving the boundary integral equation, we propose a fast algorithm for evaluating Fourier coefficients of the Newton potential by using a sparse grid approximation. When the forcing function of the Poisson equation expressed in the polar coordinates has mth-order bounded mixed derivatives, the proposed algorithm achieves an accuracy of order O(n−mlog3n), with requiring O(nlog2n) number of arithmetics for the computation, where n is the number of quadrature points used in one coordinate direction. With the help of this algorithm, the boundary integral equation derived from the Poisson equation can be efficiently solved by a fast fully discrete Fourier–Garlerkin method.

In the second part, we consider the boundary integral equation derived from Laplace equation in a domain with corners. In this case, the integral operator can be split into two operators, one is non-compact, the other is compact. We design truncation strategies for these operators' representation matrices, which compress these two dense matrices to sparse ones having only O(n) number of nonzero entries, where n is the number of the wavelet basis functions used in the method. We prove that the proposed truncation strategies do not ruin the stability and convergence rate of the integral equation. Numerical experiments are presented to verify the theoretical results and demonstrate the effectiveness of the method.

报告人简介：江颖，中山大学数据科学与计算机学院，教授，博士生导师。中山大学计算数学学士，中山大学与美国Syracuse大学联合培养博士，美国Syracuse大学数学系博士后研究员。长期从事长期从事积分方程快速算法、高维数据分析、稀疏逼近等方面的研究。主持国家自然科学基金面上项目和青年项目一项、教育部项目两项、广州市科技重点项目两项教育部项目一项，参与一项科技部重点研发计划项目，发表SCI论文20余篇。