Goal-oriented adaptive methods for a Boltzmann-type equation

Hoitinga, W.; Brummelen, van Eh Harald

doi:10.1063/1.3562629

Cited by 4 publications

(5 citation statements)

References 6 publications

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“…( 7) and ( 10)- (11). Young-Fenchel duality implies that the supremum in (19) coincides with the support function of O at x ∈ H, that is…”

Section: Worst-case Multi-objective Error Estimation Without Data Inc...mentioning

confidence: 99%

“…The worst-case multi-objective error bound assumes the conventional DWR form; cf. (10) and (19). Denoting by z ∈ H the dual solution associated with the considered supporting functional of…”

Section: Worst-case Multi-objective Error Estimation Without Data Inc...mentioning

confidence: 99%

“…It is noteworthy, however, that the use of dual solutions in a-posteriori error estimates, which forms the basis of goal-oriented error estimation, had already been pursued before by Johnson, Eriksson and Hansbo [10,21,22]. Goal-oriented adaptivity has been successfully applied to a wide variety of problems including compressible and incompressible flow problems [17,31,32], elasticity and plasticity [37,33,34], variational inequalities [38], fluid-structure interaction [45,13,36,6], free-boundary problems [46,47], phase-field models [42,44], and kinetic equations [19]. This overview is in fact far from exhaustive, and many more applications and contributions can be found in the literature.…”

Section: Introductionmentioning

confidence: 99%

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Worst-case multi-objective error estimation and adaptivity

Brummelen

Zhuk

Zwieten

2017

Computer Methods in Applied Mechanics and Engineering

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This paper introduces a new computational methodology for determining aposteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finiteelement spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multiobjective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multiobjective error in adaptive refinement procedures.

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“…( 7) and ( 10)- (11). Young-Fenchel duality implies that the supremum in (19) coincides with the support function of O at x ∈ H, that is…”

Section: Worst-case Multi-objective Error Estimation Without Data Inc...mentioning

confidence: 99%

Section: Worst-case Multi-objective Error Estimation Without Data Inc...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Worst-case multi-objective error estimation and adaptivity

Brummelen

Zhuk

Zwieten

2017

Computer Methods in Applied Mechanics and Engineering

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show abstract

“…Unweighted, residual-based estimates can be derived based on employing certain stability estimates [30], but this results in meshes independent of the choice of quantity of interest. The DWR approach has been applied to a vast number of different applications including the Poisson problem [8], nonlinear hyperbolic conservation laws [34], fluid-structure interaction problems [56], application to Boltzmann-type equations [36], as well as criticality problems in neutron transport applications [33].…”

mentioning

confidence: 99%

Linearization of the Travel Time Functional in Porous Media Flows

Houston¹,

Rourke²,

Zee³

2022

SIAM J. Sci. Comput.

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The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the post-closure safety assessment of deep geological storage facilities for radioactive waste where leaked, non-sorbing, solutes can be transported to the surface of the site by the surrounding groundwater. The accurate simulation of this transport can be attained using standard dual-weighted-residual techniques to derive goal-oriented a posteriori error bounds. This work provides a key aspect in obtaining a suitable error estimate for the travel time functional: the evaluation of its Gâteaux derivative. A mixed finite element method is implemented to approximate Darcy's equations and numerical experiments are presented to test the performance of the proposed error estimator. In particular, we consider a test case inspired by the Sellafield site located in Cumbria, in the UK.

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“…Unweighted, residual-based estimates can be derived based on employing certain stability estimates [29], but this results in meshes independent of the choice of quantity of interest. The DWR approach has been applied to a vast number of different applications including the Poisson problem [9], nonlinear hyperbolic conservation laws [33], fluid-structure interaction problems [55], application to Boltzmann-type equations [35], as well as criticality problems in neutron transport applications [32].…”

mentioning

confidence: 99%