Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation

Darrigrand, Vincent; Pardo, David; Muga, Ignacio

doi:10.1016/j.camwa.2015.03.006

Cited by 12 publications

(21 citation statements)

References 45 publications

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“…This work generalizes the classical goal-oriented procedure described in Prudhomme and Oden 34,36 by introducing an alternative operator for representing the error. We extend the results of our previous publication 38 to the multidimensional Helmholtz problem. We address the question of whether we can find an operator that provides the sharpest upper bounds independently of the approximation space.…”

Section: Discussionsupporting

confidence: 65%

“…We extend the results of our previous publication 38 to the multidimensional Helmholtz problem. We address the question of whether we can find an operator that provides the sharpest upper bounds independently of the approximation space.…”

Section: Discussionmentioning

confidence: 60%

“…In this section, we briefly recall the method described in Darrigrand et al 38 and reformulate it in terms of operators.…”

Section: Error Representations For Goal Oriented Adaptivitymentioning

confidence: 99%

“…In Section 2, we define the model Helmholtz problem used in this investigation. In Section 3, we recall the alternative method developed in Darrigrand et al, 38 and we address the issue of finding the operator that provides the sharpest error upper bounds. In Section 4, we present the p-adaptive algorithm and the software we have developed for the numerical computations.…”

Section: Introductionmentioning

confidence: 99%

“…In 1-D (see Darrigrand et al 38 ), a convergent p-adaptive algorithm was obtained either (1) by combining the classical indicators and the projection-based interpolation (PBI), [39][40][41] or (2) by using the alternative indicators (with or without PBI). This result is extended here to the 2-D and 3-D cases and applied to Helmholtz and convection-dominated diffusion problems for continuous and discontinuous coefficients.…”

Section: Introductionmentioning

confidence: 99%

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Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Darrigrand

Rodríguez-Rozas

Muga

et al. 2017

Numerical Meth Engineering

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SummaryIn goal-oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal-oriented adaptivity.While the method can be applied to a variety of problems, we focus here on two-and three-dimensional (2-D and 3-D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p-adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection-dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging-while-drilling problem to illustrate the applicability of the proposed method.

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Section: Discussionsupporting

confidence: 65%

Section: Discussionmentioning

confidence: 60%

“…In this section, we briefly recall the method described in Darrigrand et al 38 and reformulate it in terms of operators.…”

Section: Error Representations For Goal Oriented Adaptivitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Darrigrand

Rodríguez-Rozas

Muga

et al. 2017

Numerical Meth Engineering

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Forward‐in‐time goal‐oriented adaptivity

Muñoz‐Matute

Pardo

Calo

et al. 2019

Numerical Meth Engineering

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Summary In goal‐oriented adaptive algorithms for partial differential equations, we adapt the finite element mesh to reduce the error of the solution in some quantity of interest. In time‐dependent problems, this adaptive algorithm involves solving a dual problem that runs backward in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we define a pseudo‐dual problem that runs forward in time. We also describe a forward‐in‐time adaptive algorithm that works for some specific problems. Although it is not possible to define a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one‐dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. Finally, we propose a hybrid algorithm that employs the classical backward‐in‐time dual problem once and then performs the adaptive process forwards in time.

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1D Painless Multi-level Automatic Goal-Oriented h and p Adaptive Strategies Using a Pseudo-Dual Operator

Caro

Darrigrand

Álvarez-Aramberri

et al. 2022

Computational Science – ICCS 2022

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The main idea of our Goal-Oriented Adaptive (GOA) strategy is based on performing global and uniform h-or p-refinements (for h-and p-adaptivity, respectively) followed by a coarsening step, where some basis functions are removed according to their estimated importance. Many Goal-Oriented Adaptive strategies represent the error in a Quantity of Interest (QoI) in terms of the bilinear form and the solution of the direct and adjoint problems. However, this is unfeasible when solving indefinite or non-symmetric problems since symmetric and positive definite forms are needed to define the inner product that guides the refinements. In this work, we provide a Goal-Oriented Adaptive (hor p-) strategy whose error in the QoI is represented in another bilinear symmetric positive definite form than the one given by the adjoint problem. For that purpose, our Finite Element implementation employs a multi-level hierarchical data structure that imposes Dirichlet homogeneous nodes to avoid the so-called hanging nodes. We illustrate the convergence of the proposed approach for 1D Helmholtz and convectiondominated problems.

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Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation

Cited by 12 publications

References 45 publications

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Forward‐in‐time goal‐oriented adaptivity

1D Painless Multi-level Automatic Goal-Oriented h and p Adaptive Strategies Using a Pseudo-Dual Operator

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