Goal-Oriented Adaptive Mesh Refinement for Discontinuous Petrov--Galerkin Methods

Keith, Brendan; Astaneh, Ali Vaziri; Demkowicz, Leszek

doi:10.1137/18m1181754

Cited by 13 publications

(14 citation statements)

References 58 publications

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“…proving the first inequality in (54); the second inequality follows from ( 51) and the boundedness Assumption 3.…”

Section: Proposition 4 (Robustness Of the Adjoint Residual Representativementioning

confidence: 92%

“…Proof. (60) is a direct consequence of the identity (55) and the bound for the adjoint problem (54). Finally, Equations ( 61) and ( 62) are consequence of the triangular inequality…”

Section: Assumption 7 (Adjoint Saturation Condition) Let U Dgmentioning

confidence: 98%

“…Following [54], we consider the following saddle-point problem as the adjoint formulation of problem (37):…”

Section: Adjoint Saddle-point Formulationmentioning

confidence: 99%

“…This assumption ensures the existence of a localizable a posteriori error estimate in line with (54). When considering direct solvers, the third problem's solution cost is the same for both estimators, as they require to invert a matrix of the same size.…”

Section: Localizable Upper-bound Estimatesmentioning

confidence: 99%

“…Our GoA strategy is similar to a recent DPG theory [54], where they solve the adjoint problem in terms of the original saddle-point formulation with a different right-hand side. For advection-diffusion-reaction, several works explore the use of conforming FEM stabilization schemes (see [20,21,46,47,55]).…”

Section: Introductionmentioning

confidence: 99%

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Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

Rojas

Pardo

Behnoudfar

et al. 2021

Computer Methods in Applied Mechanics and Engineering

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We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goaloriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection-diffusionreaction problems.

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“…proving the first inequality in (54); the second inequality follows from ( 51) and the boundedness Assumption 3.…”

Section: Proposition 4 (Robustness Of the Adjoint Residual Representativementioning

confidence: 92%

“…Proof. (60) is a direct consequence of the identity (55) and the bound for the adjoint problem (54). Finally, Equations ( 61) and ( 62) are consequence of the triangular inequality…”

Section: Assumption 7 (Adjoint Saturation Condition) Let U Dgmentioning

confidence: 98%

“…Following [54], we consider the following saddle-point problem as the adjoint formulation of problem (37):…”

Section: Adjoint Saddle-point Formulationmentioning

confidence: 99%

Section: Localizable Upper-bound Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

Rojas

Pardo

Behnoudfar

et al. 2021

Computer Methods in Applied Mechanics and Engineering

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The DPG-star method

Demkowicz

Gopalakrishnan

Keith

2020

Computers & Mathematics with Applications

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A b s t r ac t . This article introduces the DPG-star (from now on, denoted DPG*) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG* methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG* and DPG methods can be seen as generalizations of LL * and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG* method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG* and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.

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Goal-oriented error estimation for the automatic variationally stable FE method for convection-dominated diffusion problems

Valseth

Romkes

2020

Computers & Mathematics with Applications

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Goal-Oriented Adaptive Mesh Refinement for Discontinuous Petrov--Galerkin Methods

Cited by 13 publications

References 58 publications

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

The DPG-star method

Goal-oriented error estimation for the automatic variationally stable FE method for convection-dominated diffusion problems

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