Optimal Two-level Domain Decomposition Methods for the Acoustic Wave Equation with High Wave Number

In this seminar, we will review two optimal two-level domain decomposition methods (DDMs) for the acoustic wave equation with high wave number in two and three dimensions. These two DDM preconditioners are defined over a set of overlapping subdomains, with each preconditioner formed by a global coarse solver and one local solver on each subdomain. The global coarse solver is based on the localized orthogonal decomposition (LOD) technique. The local subproblems are the acoustic problems in subdomains with homogeneous boundary conditions (for the first preconditioner) or impedance boundary conditions (for the second preconditioner). It is shown that two preconditioners are optimal, in terms of all the key parameters, such as the fine mesh size, the coarse mesh size, the subdomain size and the wave numbers. This is the first time to show the rigorous optimality of a two-level DDM with respect to all the key parameters and multiple overlapping subdomains for the acoustic wave equation with high wave number. Numerical experiments are presented to confirm the optimality and efficiency of the two proposed DDMs. This is a joint work with Peipei Lu, Xuejun Xu and and Bowen Zheng.

 

Jun Zou is currently Choh-Ming Li Chair Professor of Mathematics of The Chinese University of Hong Kong, and Head of Department of Mathematics. Before taking up his position at The Chinese University of Hong Kong in 1995, he had worked two years (93-95) in University of California at Los Angeles as a post-doctoral fellow and a CAM Assistant Professor, worked two and a half years (91-93) in Technical University of Munich as a Visiting Assistant Professor and an Alexander von Humboldt Research Fellow (Germany). He also worked two years (89-91) in Chinese Academy of Sciences (Beijing) as an Assistant Professor, right after he received his PhD degree there. Jun Zou's research interests include numerical methods and analyses of direct and inverse problems of partial differential equations. He is currently president-at-large of Hong Kong Society for Industrial and Applied Mathematics, and vice president of Hong Kong Mathematical Society. He serves currently as the associate editor of 13 international mathematics journals, including the top applied mathematics and numerical analysis journals: SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing, Journal of Scientific Computing, IMA Journal of Numerical Analysis and ESAIM: Mathematical Modelling and Numerical Analysis. Jun Zou was elected a SIAM Fellow in 2019 and an AMS Fellow in 2022.