iHDG: An Iterative HDG Framework for Partial Differential Equations

Muralikrishnan, Sriramkrishnan; Tran, Minh-Binh; Bui-Thanh, Tan

doi:10.1137/16m1074187

Cited by 14 publications

(24 citation statements)

References 45 publications

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“…From a linear algebra point of view, iHDG-I can be considered as a block Gauss-Seidel approach for the system (2) that requires only independent element-by-element and face-by-face local solves in each iteration. However, unlike conventional Gauss-Seidel schemes which are purely algebraic, the convergence of iHDG-I [45] does not depend on the ordering of unknowns. From the domain decomposition point of view, thanks to the HDG flux, iHDG can be identified as an optimal Schwarz method in which each element is a subdomain.…”

Section: Ihdg Methodsmentioning

confidence: 99%

“…From the domain decomposition point of view, thanks to the HDG flux, iHDG can be identified as an optimal Schwarz method in which each element is a subdomain. Using an energy approach, we have rigorously shown the convergence of the iHDG-I for the transport equation, the linearized shallow water equation and the convection-diffusion equation [45].…”

Section: Ihdg Methodsmentioning

confidence: 99%

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An improved iterative HDG approach for partial differential equations

Muralikrishnan

Tran

Bui-Thanh

2018

Journal of Computational Physics

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We propose and analyze an iterative high-order hybridized discontinuous Galerkin (iHDG) discretization for linear partial differential equations. We improve our previous work (SIAM J. Sci. Comput. Vol. 39, No. 5, pp. S782-S808) in several directions: 1) the improved iHDG approach converges in a finite number of iterations for the scalar transport equation; 2) it is unconditionally convergent for both the linearized shallow water system and the convection-diffusion equation;3) it has improved stability and convergence rates; 4) we uncover a relationship between the number of iterations and time stepsize, solution order, meshsize and the equation parameters. This allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize; and 5) we provide both strong and weak scalings of the improved iHDG approach up to 16, 384 cores. A connection between iHDG and time integration methods such as parareal and implicit/explicit methods are discussed. Extensive numerical results are presented to verify the theoretical findings.

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Section: Ihdg Methodsmentioning

confidence: 99%

Section: Ihdg Methodsmentioning

confidence: 99%

An improved iterative HDG approach for partial differential equations

Muralikrishnan

Tran

Bui-Thanh

2018

Journal of Computational Physics

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“…They concluded that both continuous and interior penalty discontinuous Galerkin algorithms with multigrid outperforms HDG with multigrid in terms of time to solution by a significant margin. One level Schwarz type domain decomposition algorithms in the context of HDG have been studied for elliptic equation [32,33], hyperbolic systems [36,37] and Maxwell's equations [34,35]. A balancing domain decomposition by constraints algorithm for HDG was introduced in [38] and studied for Euler and Navier-Stokes equations.…”

Section: A Brief Review On Solvers/preconditioners For Hdg Trace Systemmentioning

confidence: 99%

“…To that end, we take K = 0 in (1). Similar to [63,36], we consider the velocity field N p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 6 178 137 107 92 79 73 7 * (10 −5 ) * (10 −8 ) 167 138 113 99 8 * (10 −2 ) * (10 −2 ) * (10 −2 ) * (10 −4 ) * (10 −5 ) * (10 −7 ) Table 3: Example II: number of ML-preconditioned GMRES iterations as the mesh and solution order are refined. N p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 N p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 The solution is shown in Figure 12(a) and the difficulty of this test case comes from the presence of a curved discontinuity (shock) emanating from the inflow to the outflow boundaries.…”

Section: Example Iii: Transport Equationmentioning

confidence: 99%

A multilevel approach for trace system in HDG discretizations

Muralikrishnan¹,

Bui-Thanh²,

Shadid³

2020

Journal of Computational Physics

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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 only on the smoothness of the solution and is independent of the nature of the PDE under consideration. 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.

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“…95,96 Concerning specific solution strategies for hybrid discretization methods, a parallel solver based on the iterative Schwarz method has been developed in, 97 fast multigrid solvers have been employed in [98][99][100] and iterative approaches inspired by the Gauss-Seidel method have been discussed in. 101,102 Moreover, tailored preconditioners for the hybrid DG method have been proposed in 103,104 in the context of the Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Giacomini

Sevilla

2019

SN Appl. Sci.

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Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).

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iHDG: An Iterative HDG Framework for Partial Differential Equations

Cited by 14 publications

References 45 publications

An improved iterative HDG approach for partial differential equations

An improved iterative HDG approach for partial differential equations

A multilevel approach for trace system in HDG discretizations

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

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