Convergence of natural adaptive least squares finite element methods

Carstensen, Carsten; Park, Eun Jae; Bringmann, Philipp

doi:10.1007/s00211-017-0866-x

Cited by 16 publications

(8 citation statements)

References 24 publications

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“…From this, we then conclude that the sequence of discrete solutions produced by the above iteration converges to the exact solution of the underlying problem (see Theorem 2 below). Let us mention that our result is weaker than [15,Theorem 4.1] as we only show (plain) convergence whereas [15] proves that an equivalent error quantity is contractive. However, the latter result comes at the price of assuming sufficiently fine initial meshes and sufficiently large marking parameters 0 < θ < 1 in the Dörfler marking criterion.…”

Section: Introductioncontrasting

confidence: 60%

“…The latter contradicts somehow the by now standard proofs of optimal convergence where θ needs to be sufficiently small; see [13] for an overview. However, we also note that [15] is constrained by the Dörfler marking criterion, while the present analysis, in the spirit of [33], covers a fairly wide range of marking strategies.…”

Section: Introductionmentioning

confidence: 85%

“…In contrast to this, only very little is known about convergence of adaptive LSFEM; see [12,8,14,15]. To the best of our knowledge, plain convergence of adaptive LSFEM when using the built-in error estimator has so far only been addressed in [15]. The other works [12,8,14] deal with optimal convergence results, but rely on alternative error estimators.…”

Section: Introductionmentioning

confidence: 99%

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A short note on plain convergence of adaptive least-squares finite element methods

Führer,

Praetorius

2020

Preprint

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We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, meshrefinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract framework which covers a variety of model problems.

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Section: Introductioncontrasting

confidence: 60%

Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

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A short note on plain convergence of adaptive least-squares finite element methods

Führer,

Praetorius

2020

Preprint

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“…Nevertheless, many publications and the experiments from the preceding section show the convergence of the natural adaptive dPG methods employing the built-in error estimator. The proof of optimal convergence rates of this natural adaptive dPG algorithm is an open problem, even in the context of the more extensively studied least-squares methods, where there is only a result on plain convergence [CPB17].…”

Section: Discussionmentioning

confidence: 99%

Optimal Convergence Rates for Adaptive Lowest-Order Discontinuous Petrov--Galerkin Schemes

Carstensen¹,

Hellwig²

2018

SIAM J. Numer. Anal.

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The thesis »Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods« proves optimal convergence rates for four lowest-order discontinuous Petrov-Galerkin methods for the Poisson model problem for a sufficiently small initial mesh-size in two different ways by equivalences to two other non-standard classes of finite element methods, the reduced mixed and the weighted Least-Squares method. The first is a mixed system of equations with first-order conforming Courant and nonconforming Crouzeix-Raviart functions. The second is a generalized Least-Squares formulation with a midpoint quadrature rule and weight functions. The thesis generalizes a result on the primal discontinuous Petrov-Galerkin method from [Carstensen, Bringmann, Hellwig, Wriggers 2018] and characterizes all four discontinuous Petrov-Galerkin methods simultaneously as particular instances of these methods. It establishes alternative reliable and efficient error estimators for both methods. A main accomplishment of this thesis is the proof of optimal convergence rates of the adaptive schemes in the axiomatic framework [Carstensen, Feischl, Page, Praetorius 2014]. The optimal convergence rates of the four discontinuous Petrov-Galerkin methods then follow as special cases from this rate-optimality. Numerical experiments verify the optimal convergence rates of both types of methods for different choices of parameters. Moreover, they complement the theory by a thorough comparison of both methods among each other and with their equivalent discontinuous Petrov-Galerkin schemes. ZusammenfassungDie vorliegende Arbeit »Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods« beweist optimale Konvergenzraten für vier diskontinuierliche Petrov-Galerkin (dPG) Finite-Elemente-Methoden für das Poisson-Modell-Problem für genügend feine Anfangstriangulierung. Sie zeigt dazu die Äquivalenz dieser vier Methoden zu zwei anderen Klassen von Methoden, den reduzierten gemischten Methoden und den verallgemeinerten Least-Squares-Methoden. Die erste Klasse benutzt ein gemischtes System aus konformen Courantund nichtkonformen Crouzeix-Raviart-Finite-Elemente-Funktionen. Die zweite Klasse verallgemeinert die Standard-Least-Squares-Methoden durch eine Mittelpunktsquadratur und Gewichtsfunktionen. Diese Arbeit verallgemeinert ein Resultat aus [Carstensen, Bringmann, Hellwig, Wriggers 2018], indem die vier dPG-Methoden simultan als Spezialfälle dieser zwei Klassen charakterisiert werden. Sie entwickelt alternative Fehlerschätzer für beide Methoden und beweist deren Zuverlässigkeit und Effizienz. Ein Hauptresultat der Arbeit ist der Beweis optimaler Konvergenzraten der adaptiven Methoden durch Beweis der Axiome aus [Carstensen, Feischl, Page, Praetorius 2014]. Daraus folgen dann insbesondere die optimalen Konvergenzraten der vier dPG-Methoden. Numerische Experimente bestätigen diese optimalen Konvergenzraten für beide Klassen von Methoden. Außerdem ergänzen sie die Theorie durch ausführliche Vergleiche beider Methoden untereinander und mit den äquivalenten dPG-Methoden. C...

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“…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.…”

Section: Towards Adaptivitymentioning

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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Convergence of natural adaptive least squares finite element methods

Cited by 16 publications

References 24 publications

A short note on plain convergence of adaptive least-squares finite element methods

A short note on plain convergence of adaptive least-squares finite element methods

Optimal Convergence Rates for Adaptive Lowest-Order Discontinuous Petrov--Galerkin Schemes

Towards adaptive discontinuous Petrov‐Galerkin methods

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