Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem

Ahmed, Naveed; Becher, Simon; Matthies, Gunar

doi:10.1016/j.cma.2016.09.026

Cited by 29 publications

(33 citation statements)

References 30 publications

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“…Since recently, variational time discretization schemes based on continuous or discontinuous finite element techniques have been developed to the point that they can be put into use (cf., e.g., [1,2,6,13,18,17,23,15] and the references therein) and demonstrated their significant advantages. Higher order methods are naturally embedded in these schemes and the uniform variational approach simplifies stability and error analyses.…”

Section: Introduction and Mathematical Modelmentioning

confidence: 96%

“…, m = 2, s = 2, τ n = 0.005 dG(1), m = 2, s = 2, τ n = 0.010 cG(1), m = 2, s = 2, τ n = 0.010 dG(1), m = 2, s = 2, τ n = 0.020 Total fixed-stress iterations for varying step length size τ n for dG(1) in time. , m = 2, s = 2, τ n = 0.005 cGP(2), m = 2, s = 2, τ n = 0.010 cGP(1), m = 2, s = 2, τ n = 0.010 cGP(2), m = 2, s = 2, τ n = 0.020 Total fixed-stress iterations for varying step length size τ n for cGP(2) in time.…”

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confidence: 99%

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Space–time finite element approximation of the Biot poroelasticity system with iterative coupling

Bause

Radu

Köcher

2017

Computer Methods in Applied Mechanics and Engineering

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We analyze an optimized artificial fixed-stress iteration scheme for the numerical approximation of the Biot system modelling fluid flow in deformable porous media. The iteration is based on a prescribed constant artificial volumetric mean total stress in the first half step. The optimization comes through the adaptation of a numerical stabilization or tuning parameter and aims at an acceleration of the iterations. The separated subproblems of fluid flow, written as a mixed first order in space system, and mechanical deformation are discretized by space-time finite element methods of arbitrary order. Continuous and discontinuous discretizations of the time variable are encountered. The convergence of the iteration schemes is proved for the continuous and fully discrete case. The choice of the optimization parameter is identified in the proofs of convergence of the iterations. The analyses are illustrated and confirmed by numerical experiments.

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Section: Introduction and Mathematical Modelmentioning

confidence: 96%

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confidence: 99%

Space–time finite element approximation of the Biot poroelasticity system with iterative coupling

Bause

Radu

Köcher

2017

Computer Methods in Applied Mechanics and Engineering

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“…Let D be a Lipschitz domain in R 3 . We consider the time-harmonic version of Maxwell's equations in the low-frequency regime where the displacement currents are negligible; see §43.1.…”

Section: Maxwell's Equationsmentioning

confidence: 99%

“…Finally, we use the same arguments as in the proof of Theorem 73.8 to bound δ hτ ℓ 2 (J;L 2 ) with δ hτ := (δ n h ) n∈Nτ and (73.22b) follows by invoking the triangle inequality. Examples include continuous interior penalty as in Burman and Fernández [65], local projection stabilization as in Arndt et al [14], Dallmann et al [99], Ahmed et al [3,4], and subgrid viscosity as in Guermond et al [164]. Many other techniques can be used as well (see for instance Codina [90]), and the literature is prolific on the subject.…”

Section: Error Analysismentioning

confidence: 99%

Finite Elements III

Ern

Guermond

2021

Texts in Applied Mathematics

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In Part XII, composed of Chapters 56 to 63, we study the finite element approximation of PDEs where a coercivity property is not available, so that the analysis solely relies on inf-sup conditions. Stability can be obtained by employing various stabilization techniques (residual-based or fluctuation-based). In the present chapter, we introduce the prototypical model problem we are going to work on: it is a system of first-order linear PDEs introduced in 1958 by Friedrichs [131]. This system enjoys symmetry and positivity properties and is often referred to in the literature as Friedrichs' system. Friedrichs wanted to handle within a single functional framework PDEs that are partly elliptic and partly hyperbolic, and for this purpose he developed a formalism that goes beyond the traditional classification of PDEs into elliptic, parabolic, and hyperbolic types. Friedrichs' formalism is very powerful and encompasses several model problems. Important examples are the advection-reaction equation, the div-grad problem related to Darcy's equations, and the curl-curl problem related to Maxwell's equations. This theory will be used systematically in the following chapters. All the theoretical arguments in this chapter are presented assuming that the functions are complex-valued. The real-valued case can be obtained by replacing the field C by R, by replacing the Hermitian transpose Z H by the transpose Z T , and by removing the real part symbol ℜ. Basic ideasLet D be a Lipschitz domain in R d . We consider functions defined on D with values in C m for some integer m ≥ 1. The (Hermitian) inner product inWe denote by I m the identity matrix in C m×m .M , we observe that u − I h0 (u) ∈ V 0 . Using (56.26), we infer that for all v ∈ V 0 = ker(M − N ),with φ := max(ℓ * , µ 0 h). If the boundary of D is not smooth and/or if the coefficients σ, µ are discontinuous, it is in general preferable to use H(curl)-conforming finite element subspaces based on edge elements (see Chapter 15) instead of using H 1 -conforming finite element subspaces; see §45.4.

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“…The postprocessing technique applied to a numerical solution increases the global smoothness by one differentiation order and the local polynomial degree by one. This results in improved accuracy in integral-based norms, see [1,3,4,6,8,14,24]. Note that the postprocessing comes with almost no computational costs since just jumps of derivatives of the discrete solution are needed.…”

Section: Introductionmentioning

confidence: 99%

Variational time discretizations of higher order and higher regularity

Becher

Matthies

2021

Bit Numer Math

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We consider a family of variational time discretizations that are generalizations of discontinuous Galerkin (dG) and continuous Galerkin–Petrov (cGP) methods. The family is characterized by two parameters. One describes the polynomial ansatz order while the other one is associated with the global smoothness that is ensured by higher order collocation conditions at both ends of the subintervals. Applied to Dahlquist’s stability problem, the presented methods provide the same stability properties as dG or cGP methods. Provided that suitable quadrature rules of Hermite type are used to evaluate the integrals in the variational conditions, the variational time discretization methods are connected to special collocation methods. For this case, we present error estimates, numerical experiments, and a computationally cheap postprocessing that allows to increase both the accuracy and the global smoothness by one order.

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Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem

Cited by 29 publications

References 30 publications

Space–time finite element approximation of the Biot poroelasticity system with iterative coupling

Space–time finite element approximation of the Biot poroelasticity system with iterative coupling

Finite Elements III

Variational time discretizations of higher order and higher regularity

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