A low-order discontinuous Petrov–Galerkin method for the Stokes equations

Carstensen, Carsten; Puttkammer, Sophie

doi:10.1007/s00211-018-0965-3

Cited by 7 publications

(4 citation statements)

References 24 publications

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“…This is already a new result even for the linear cases in [9,13] and opens the door of a convergence analysis of adaptive algorithms via a generalization of [11,14]. This paper contributes the aforementioned equivalent characterizations and a first convergence analysis in the natural norms.…”

Section: Introductionmentioning

confidence: 79%

“…It has been discussed in [5,9,13] that the norm of the computed residual }y h } Y " }F´bpv C , q RT ; ‚ q} Yh is almost a computable error estimator for linear problems and this paper extends it to the a posteriori error estimate…”

Section: Introductionmentioning

confidence: 99%

“…To prove the efficiency, utilize F " Bpxq, Lemma 2.9, and (2.12) to verify}F ´Bpx h q} Y ˚" }Bpxq ´Bpx h q} Y Remark 2 13. Since y " 0 and y h is computed, the a posteriori error }y ´yh } Y " }y h } Y is already an error estimator and can be added on both sides of the reliability (resp.…”

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confidence: 99%

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Nonlinear discontinuous Petrov–Galerkin methods

et al. 2018

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The discontinuous Petrov-Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation, a reduced formulation, and a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework for the approximation of a regular solution. The variational model example allows for a built-in guaranteed error control despite inexact solve. The subtle uniqueness of discrete minimizers is monitored in numerical examples. Classification (2000) 47H05,49M15,65N12,65N15,65N30 Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis' under the project 'Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics' (CA Mathematics Subject

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Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

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Nonlinear discontinuous Petrov–Galerkin methods

et al. 2018

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“…The first low-order dPG method reduces these costs for the Poisson model problem [14]. A low-order approach for the Stokes equations based on an ultra-weak form of the pseudo stress formulation is introduced in [15] and allows for the computation of guaranteed error bound in terms of the Friedrichs and the tr-dev-div constants. Numerical experiments suggest that adaptive refinement driven by the residual error estimator is successful.…”

Section: Applicationsmentioning

confidence: 99%

Towards adaptive discontinuous Petrov‐Galerkin methods

Bringmann

Carstensen

Gallistl

et al. 2016

Proc Appl Math and Mech

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The discontinuous Petrov-Galerkin (dPG) method is a minimum residual method with broken test functions for instant stability. The methodology is written in an abstract framework with product spaces. It is applied to the Poisson model problem, the Stokes equation, and linear elasticity with low-order discretizations. The computable residuum leads to guaranteed error bounds and motivates adaptive refinements. FrameworkThe dPG paradigm suggests some spatial decomposition of test functions in a framework of a minimal residual method with a partition T , the Hilbert space Y := K∈T Y (K) and the seminormed vector spaceX := K∈T X(K). Suppose that the bounded bilinear form b :X × Y → R is nondegenerate for all elements of the closed normed ansatz space X ⊂X in the sense thatGiven F ∈ Y * and subspaces X h ⊆ X, Y h ⊆ Y , the dPG method approximates the solution u ∈ X to the variational problemIf the discrete inf-sup condition holds or, equivalently [2], if there exists a linear bounded projector P : Y → Y onto Y h with norm P such that the annulation property b(x h , y − P y) = 0 holds for all x h ∈ X h , y ∈ Y , then the mixed problem (M h ) is well-posed and best-approximation holds with [3]Furthermore, the annulation operator P leads to the efficient and reliable a posteriori error control [4] Towards AdaptivityThe recent progress in the analysis of minimal residual methods with adaptive mesh refinement might lead to the quasi-optimal convergence of adaptive dPG methods as well. The related least-squares FEMs seek discrete minimizers of a least-squares functional LS(f ; •) whose element-wise evaluation yields a reliable and efficient built-in a posteriori error estimator. The plain convergence of this natural adaptive least-squares FEM is proven by [5]. The standard techniques for quasi-optimal convergence proofs [6-8] cannot be applied in this context due to the lack of the reduction property of the minimal residual functional and an additional data approximation term which needs to be reduced. As a remedy, a separate marking algorithm can guarantee the reduction of an alternative a posteriori error η(T , •) and the data approximation error f − Π 0 f L 2 (Ω) with the piecewise constant L 2 best-approximation Π 0 f of f ∈ L 2 (Ω) and enables the proof of quasi-optimal convergence [9]. This result for the Poisson model problem is generalized to the Stokes equations [10] and linear elasticity [11]. All these proofs base on the framework of the axioms of adaptivity [8] which is generalized to separate marking algorithms by [12].

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