CAREER: Numerical Analysis for Meshfree and Particle Methods via Nonlocal Models

Meshfree and particle methods are widely used in the computational studies of partial differential equations (PDEs) with applications to many mechanical, hydrodynamical and biophysical processes. They offer many advantages over traditional mesh-based methods, particularly for problems with complex or moving geometries, large deformations of materials, or other singular behaviors of solutions. The goal of this project is to investigate an innovative approach for designing and analyzing meshfree and particle methods through the study of continuum nonlocal models and their robust discretizations. A central idea is to design "asymptotically compatible schemes" with respect to the nonlocal/integral relaxation scale of PDEs so as to advance the stability, accuracy and efficiency of meshfree and particle methods. The proposed research will advance the theoretical understanding of meshfree and particle methods and enhance their practical performance and functionality. Central to the project is an integrated plan of educational goals, and these include promoting the engagement and retention of female and underrepresented groups in math and science, enhancing applied and computational mathematics curricula, and disseminating new scientific discoveries. The PI proposes a variety of activities that aim at promoting computational thinking in middle/high school students and teachers, inspiring and preparing undergraduates for early research experience, and contributing to a more connected and inclusive intellectual environment. The project primarily focuses on three objectives. The first is the development of monotone meshfree methods for elliptic equations via nonlocal relaxation. This includes the study of linear and nonlinear elliptic equations in Euclidean space and on manifolds. The central theme is to preserve the monotonicity in the discretization while keeping a compact selection of nearby points. The second subject is meshfree methods for systems of equations via the study of nonlocal vector calculus. The study of well-posed nonlocal models through nonlocal vector calculus is crucial for designing of stable numerical discretizations, without which instabilities are often observed (e.g., in the SPH method and peridynamics correspondence model). In the last subproject, we focus on improving the accuracy and efficiency of Lagrange type particle methods for nonlinear time-dependent PDEs through nonlocal analysis. The discussion includes aggregation equation, degenerate diffusion, and convection-diffusion equation. The study of Wasserstein gradient flows will also be needed for rigorous convergence studies. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.