NDLI: iHDG: An Iterative HDG Framework for Partial Differential Equations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Muralikrishnan, Sriramkrishnan Tran, Minh-Binh Bui-Thanh, Tan
Copyright Year	2017
Abstract	We present a scalable iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of linear partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations and hence inherits advances from both sides. In particular, the method can be viewed as a Gauss--Seidel approach that requires only independent element-by-element and face-by-face local solves in each iteration. As such, it is well suited for current and future computing systems with massive concurrencies. Unlike conventional Gauss--Seidel schemes which are purely algebraic, the convergence of iHDG, thanks to the built-in HDG numerical flux, does not depend on the ordering of unknowns. We rigorously show the convergence of the proposed method for the transport equation, the linearized shallow water equation, and the convection-diffusion equation. For the transport equation, the method is convergent regardless of mesh size $h$ and solution order $p$, and furthermore the convergence rate is independent of the solution order. For the linearized shallow water and the convection-diffusion equations we show that the convergence is conditional on both $h$ and $p$. Extensive steady and time-dependent numerical results for the 2D and 3D transport equations, the linearized shallow water equation, and the convection-diffusion equation are presented to verify the theoretical findings.
Sponsorship	U.S. Department of Energy. H2020 European Research Council. National Science Foundation
Starting Page	S782
Ending Page	S808
Page Count	27
File Format	PDF
ISSN	10648275
DOI	10.1137/16M1074187
e-ISSN	10957197
Issue Number	5
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-10-26
Access Restriction	Subscribed
Subject Keyword	convection-diffusion equation Stability and convergence of numerical methods Solution of discretized equations Schwarz methods Iterative methods for linear systems iterative solvers hybridized discontinuous Galerkin methods Gauss--Seidel methods shallow water equation Multigrid methods; domain decomposition Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods transport equation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in