Block Jacobi for Discontinuous Galerkin Discretizations: No Ordinary Schwarz Methods

Gander, Martin J.; Hajian, Soheil

doi:10.1007/978-3-319-05789-7_27

Cited by 7 publications

(5 citation statements)

References 5 publications

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“…We also make connection between our proposed iHDG-II approach with parareal and time integration methods. Last but not least, our framework is more general: indeed it recovers the contraction factor results in [46] for elliptic equations as one of the special cases.…”

Section: Ihdg Methodssupporting

confidence: 53%

“…This is due to the new way of computing the weighted trace in (5) that involves u k+1 , and hence changing the structure of the local solves. Similar and independent work for HDG methods for elliptic/parabolic problems have appeared in [46,39,47]. Here, we are interested in pure hyperbolic equations/systems and convectiondiffusion equations.…”

Section: Ihdg Methodsmentioning

confidence: 96%

“…We have developed and analyzed one such solver namely iterative HDG (iHDG) in our previous work [45] for elliptic, scalar and system of hyperbolic equations. Independent and similar efforts for elliptic and parabolic equations have been proposed and analyzed in [46,39,47].…”

Section: Introductionmentioning

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 Methodssupporting

confidence: 53%

Section: Ihdg Methodsmentioning

confidence: 96%

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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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“…The optimized Schwarz method is introduced by Lions [8], which is based on Robin interface condition and can be applied both overlapping and non-overlapping cases. In [6,4,5], the optimized Schwarz method of the HDG method is proposed and analyzed. The Neumann-Neumann method [9] and the FETI (or Dirichlet-Dirichlet) method [3] are also well known as a nonoverlapping algorithm, however, there is no application to the HDG method to the best of our knowledge.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Reduced-Order HDG Method for the Stokes Equations

Oikawa

2015

J Sci Comput

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Abstract.In this paper, we analyze a hybridized discontinuous Galerkin(HDG) method with reduced stabilization for the Stokes equations. The reduced stabilization enables us to reduce the number of facet unknowns and improve the computational efficiency of the method. We provide optimal error estimates in an energy and L 2 norms. It is shown that the reduced method with the lowestorder approximation is closely related to the nonconforming Crouzeix-Raviart finite element method. We also prove that the solution of the reduced method converges to the nonconforming Gauss-Legendre finite element solution as a stabilization parameter τ tends to infinity and that the convergence rate is O(τ −1 ).

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“…In the classical finite element case a block Jacobi relaxation is equivalent to a classical Schwarz method with Dirichlet transmission conditions, see for example [13]. This is however not necessarily the case for discontinuous Galerkin methods, see [14]. We investigate in this short paper what kind of domain decomposition methods one obtains when simply performing a block Jacobi relaxation in a PWDG discretization of the Helmholtz equation, and also show how one can obtain optimized Schwarz methods for such discretizations.…”

Section: Introductionmentioning

confidence: 99%