A-stable discontinuous Galerkin–Petrov time discretization of higher order

Schieweck, Friedhelm

doi:10.1515/jnum.2010.002

Cited by 49 publications

(52 citation statements)

References 7 publications

(10 reference statements)

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“…However, the costs of one multigrid iteration in the cGP(2) or dG(1) method is almost 3 times higher than in cGP (1). Nevertheless, for a desired accuracy of 10 7 , the cGP(2) scheme is about 5 times faster than cGP(1) due to the much larger time step size required for cGP (2).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Next, in each time step, we apply a standard Galerkin finite element discretization with the so-called Q 2 …”

Section: Space Discretization By Femmentioning

confidence: 99%

“…These methods are of higher order due to their theoretical construction. Moreover, the cGP-method is Astable [2] and the dG-method is L-stable [3] for each polynomial degree k in time. In our recent paper [1], we described both methods in detail for the heat equation and proposed an efficient multigrid method for solving the according block systems in each time step.…”

Section: Introductionmentioning

confidence: 99%

“…They have proved optimal error estimates as well as superconvergence results at the discrete points t n of the time mesh. In [2], the cGP-method was presented as a ''discontinuous Galerkin Petrov'' method where the name was chosen due to the fact that the test space is discontinuous in time and different from the ansatz space. The method was analyzed for the linear case in an abstract Hilbert space and for the non-linear case in the Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

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A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

Hussain¹

2012

TONUMJ

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Abstract:In this note, we extend our recent work for the heat equation in [1] and describe and compare by means of numerical experiments the continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time discretization applied to the nonstationary Stokes equations in the two-dimensional case. For the space discretization, we use the well-known LBB-stable quadrilateral finite element which consists of conforming biquadratic elements for the velocity and discontinuous linear elements for the pressure. We discuss implementation aspects as well as methods for solving the resulting block systems using monolithic multigrid solvers based on Vanka-type smoothers. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs. We show that the convergence behavior of the multigrid method is almost independent of the mesh size in space and the time step size which means (at least for these examples) that we have created an efficient solution process. Mathematics Subject Classification (MSC):65M12, 65M55, 65M60.

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Section: Numerical Resultsmentioning

confidence: 99%

“…Next, in each time step, we apply a standard Galerkin finite element discretization with the so-called Q 2 …”

Section: Space Discretization By Femmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

Hussain¹

2012

TONUMJ

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“…In the present paper, we compare this approach, which we call continuous Galerkin-Petrov discretization (cGP(k)-method, [4]), with the well-known discontinuous Galerkin time discretization (dG(k)-method, [5]). For the cGP(k)-method, the discrete solution space consists of continuous piecewise polynomial functions in time of degree k 1 and the discrete test space of discontinuous polynomial functions of degree k − 1.…”

Section: Introductionmentioning

confidence: 99%

Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation

Hussain

Schieweck

Turek

2011

Journal of Numerical Mathematics

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We discuss numerical properties of continuous Galerkin-Petrov and discontinuous Galerkin time discretizations applied to the heat equation as a prototypical example for scalar parabolic partial differential equations. For the space discretization, we use biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss implementation aspects of the time discretization as well as efficient methods for solving the resulting block systems. Here, we compare a preconditioned BiCGStab solver as a Krylov space method with an adapted geometrical multigrid solver. Only the convergence of the multigrid method is almost independent of the mesh size and the time step leading to an efficient solution process. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs.

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An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

Hussain

Schieweck

Turek

2013

Numerical Methods in Fluids

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SUMMARYIn this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time‐stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well‐known LBB‐stable finite element pair Q2MathClass-bin∕P1disc of third‐order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka‐like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.

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A-stable discontinuous Galerkin–Petrov time discretization of higher order

Cited by 49 publications

References 7 publications

A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

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