Enhancing Meshfree Methods for Solving PDEs through Nonlocal Analysis

Abstract: Meshfree and particle methods are widely used in computational studies of partial differential equations, offering many advantages compared to traditional mesh- or grid-based numerical methods.  Nevertheless, many practical questions revolve around meshfree methods, encompassing concerns about stability and accuracy.  We propose a new paradigm of designing meshfree methods for solving partial differential equations (PDEs) through nonlocal analysis, inspired by the recent development of nonlocal calculus and its applications.  We propose that the development of stable, accurate, and efficient meshfree methods relies on two key factors: (1) the formulation of well-posed continuum nonlocal models to approximate the PDE models and (2) the development of asymptotically compatible schemes for robust discretization of nonlocal models that allow a flexible coupling of the modeling and discretization parameters.  We will review several aspects of nonlocal calculus and asymptotically compatible schemes and demonstrate the idea with a monotone meshfree method for solving linear elliptic equations in non-divergence form.

Speaker’s Bio: Xiaochuan Tian is an Assistant Professor of Mathematics at the University of California, San Diego.  She received her Ph.D. from Columbia University and was a Bing Instructor in the Department of Mathematics at the University of Texas, Austin.  Her areas of research interest include numerical PDEs, nonlocal integral models, multiscale and stochastic modeling, and most recently, intersections of PDEs and data analysis.  Her research is partially funded by the National Science Foundation CAREER grant and the Alfred P. Sloan Fellowship.