報告題目：GoFD and Meshfree GoFD method for the fractional Laplacian on arbitrary bounded domains
報 告 人： 沈金葉 副教授 西南财經大學
報告時間：2024年4月16日 9:30-10:30
報告地點：#騰訊會議：268-515-977，會議密碼：20857
校内聯系人： wxjldx@jlu.edu.cn
報告摘要：A grid-overlay finite difference method (GoFD) was proposed and developed for the numerical solution of homogeneous Dirichlet boundary value problems of the fractional Laplacian on arbitrary bounded domains. It was shown to have advantages of both finite difference and finite element methods, including its efficient implementation through the fast Fourier transform and ability to work for complex domains and with mesh adaptation. The purpose of this talk is to introduce GoFD and GoFD in a meshfree setting. GoFD method uses an unstructured simplicial mesh and an overlay uniform grid for the underlying domainand constructs the approximation based on a uniform-grid finite difference approximation and a data transfer from the unstructured mesh to the uniform grid. It is shown that its stiffness matrix is similar to a symmetric and positive definite matrix and thus invertible if the data transfer has full column rank and positive column sums. Moreover, a key of meshfree GoFD is to construct the data transfer matrix from a given point cloud to a uniform grid. Two approaches are proposed, one based on the moving least squares fitting and the other based on the Delaunay triangulation and piecewise linear interpolation. Numerical results obtained for examples with convex and concave domains and various types of point clouds are presented.
報告人簡介： 沈金葉，副教授，西南财經大學伟德线上平台碩士生導師。研究興趣：金融期權定價模型的數值算法，Bernoulli 自由邊界問題的自适應算法，非線性發展方程的差分方法，分數階模型的數值算法。主持國家自然科學基金青年基金項目1項，參與完成國家自然科學基金面上項目2項。近5年在國際主流計算數學，金融數學雜志上發表學術論文20餘篇。長期從事《數值分析》《偏微分方程數值解》等課程的教學工作。
