Discontinuous Galerkin approximations for elliptic problems

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.

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“…We also recall the following result given [13,14] and, for the sake of completeness, outline a sketch of the proof.…”

Section: Main Propertiesmentioning

confidence: 99%

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

Antonietti

Houston

2010

J Sci Comput

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“…(4) F denotes the sum of the convective and viscous flux functions, and F the sum of their numerical counterparts. For the former the flux computation is based on the exact solution of the Riemann problem for the artificial compressibility perturbation of the locally 1D inviscid Euler equations, as suggested in [7,8], while for the latter the BR2 scheme is employed, proposed in [9] and theoretically analyzed in [11,1].…”

Section: Dg Space Discretizationmentioning

confidence: 99%

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

Massa¹,

Noventa²,

Bassi³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. In recent years the increasing attention to high-order Finite Volume (FV), Finite Element (FE) and spectral methods and the growth of computing power promote the development of high-order temporal schemes to perform robust, accurate and efficient unsteady long-time simulations. In this context, some features of the Discontinuous Galerkin finite element (DG) methods, e.g. compactness and flexibility, can be advantageous both for explicit and implicit time integration approaches. Explicit schemes can achieve very high accuracy, but are limited by time-step restrictions, while implicit schemes, even if memory consuming, can overcome time-step limitations, thus improving the time integration efficiency. During last decades several high order implicit temporal schemes have been developed, and some of them have been successfully coupled with DG methods. However these schemes can show the order reduction if applied to very stiff problems or problems with time-dependent boundary conditions. To overcome these limitations, high-order linearly implicit two-step peer methods have been proposed and successfully applied to the numerical solution of differential-algebraic equations. The aim of the present work is to implement high-order two-step peer methods in a DG code and assess their performance for the unsteady solution of the incompressible Navier-Stokes (INS) and Reynolds Averaged Navier-Stokes equations (RANS) equations.

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“…(Sketchy) Here we demonstrate only the terms that are involved in estimations of the interelement jump terms, which are additional to those in the SD-case. To this end, we introduce R : W h → W d , see [4], defined by …”

Section: Theorem 4 Under the Assumptions Of Theorem 3 And For The Exmentioning

confidence: 99%

“…As for (P1) we employ streamline-diffusion and discontinuous Galerkin methods based on the studies in [8] and [4]. We shall only give sketch of the proofs.…”

Section: Introductionmentioning

confidence: 99%

A combined discontinuous Galerkin and finite volume scheme for multi-dimensional VPFP system

Asadzadeh¹,

Bartoszek²

2011

AIP Conference Proceedings

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Discontinuous Galerkin approximations for elliptic problems

Abstract: In this paper we analyze a discontinuous finite element method recently introduced by Bassi and Rebay for the approximation of elliptic problems. Stability and error estimates in various norms are proven.

Cited by 322 publications

References 13 publications

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

A combined discontinuous Galerkin and finite volume scheme for multi-dimensional VPFP system

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