A Novel and Simple Spectral Method for Nonlocal PDEs with the Fractional Laplacian

Abstract: In this talk, we mainly introduce a novel and simple spectral method to discretize the fractional Laplacian and apply it to study nonlocal elliptic problems.  The key idea of our method is to apply the semi-discrete Fourier transforms to approximate the pseudo-differential definition of the fractional Laplacian.  Our scheme can be viewed as a discrete pseudo-differential operator with symbol |\xi|^\alpha,  and thus it provides an exact discrete analogue of the fractional Laplacian.  This new method is different from the Fourier pseudospectral methods in the literature, which are usually limited to periodic boundary conditions.  We provide both numerical analysis and experiments to study the performance of our method.  It shows that our method has a spectral accuracy if function u is smooth enough.  Detailed numerical analysis is also presented under different smoothness conditions.  Moreover, the stability and convergence of our method in solving the fractional Poisson equations were analyzed.  Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector products.  The computational complexity is O(N log N), and the memory storage is O(N) with N the total number of points.  This is a joint work with Shiping Zhou (Missouri University of Science and Technology).

Speaker’s Bio: Dr. Yanzhi Zhang is a professor of computational and applied mathematics in the Department of Mathematics and Statistics at Missouri University of Science and Technology (Missouri S&T).  She received her Ph.D. in Computational Mathematics from the National University of Singapore in 2006, worked as a postdoc in the Department of Scientific Computing at Florida State University, and joined the faculty of Missouri S&T in 2010.  Dr. Zhang’s current research interests include data-informed modeling for seismic waves, machine learning algorithms and applications, fractional PDEs and nonlocal models, and multiscale/multiphysics modeling and simulations.