iHDG: an iterative HDG framework for partial differential equations. 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.
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References in zbMATH (referenced in 8 articles , 1 standard article )
Showing results 1 to 8 of 8.
- Chen, Gang; Monk, Peter; Zhang, Yangwen: Superconvergent HDG methods for Maxwell’s equations via the (M)-decomposition (2022)
- Rhebergen, Sander; Wells, Garth N.: Preconditioning for a pressure-robust HDG discretization of the Stokes equations (2022)
- Kang, Shinhoo; Giraldo, Francis X.; Bui-Thanh, Tan: IMEX HDG-DG: a coupled implicit hybridized discontinuous Galerkin and explicit discontinuous Galerkin approach for shallow water systems (2020)
- Muralikrishnan, Sriramkrishnan; Bui-Thanh, Tan; Shadid, John N.: A multilevel approach for trace system in HDG discretizations (2020)
- Shen, Jiguang; Singler, John R.; Zhang, Yangwen: HDG-POD reduced order model of the heat equation (2019)
- Muralikrishnan, Sriramkrishnan; Tran, Minh-Binh; Bui-Thanh, Tan: An improved iterative HDG approach for partial differential equations (2018)
- Muralikrishnan, Sriramkrishnan; Tran, Minh-Binh; Bui-Thanh, Tan: iHDG: an iterative HDG framework for partial differential equations (2017)
- Roberts, Nathan V.; Chan, Jesse: A geometric multigrid preconditioning strategy for DPG system matrices (2017)