A multilevel approach for trace system in HDG discretizations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:407

Article

Muralikrishnan, Sriramkrishnan¹  Tan Bui-Thanh^1,2  Shadid, John N.^3,4 

[1] Univ Texas Austin, Dept Aerosp Engn & Engn Mech, Austin, TX 78712 USA

[2] Univ Texas Austin, Oden Inst Computat Engn & Sci, Austin, TX 78712 USA

[3] Sandia Natl Labs, Computat Math Dept, POB 5800,MS 1321, Albuquerque, NM 87185 USA

[4] Univ New Mexico, Dept Math & Stat, Albuquerque, NM 87131 USA

关键词: Iterative solvers;    Multilevel solvers;    Hybridized discontinuous Galerkin methods;    Transport equation;    Convection-diffusion equation;    Nested dissection;   

DOI  :  10.1016/j.jcp.2020.109240

来源: Elsevier

【 摘 要 】

We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly h-coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends on the smoothness of the solution and can offer significant speedups and memory savings compared to the nested dissection direct solver. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms with respect to solution order. While scalability with mesh size in general is not guaranteed and depends on the smoothness of the solution and the type of equation, improving it is a part of future work. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free