Stabilized finite element approximations for a generalized Boussinesq problem: A posteriori error analysis

Allendes, Alejandro; Naranjo, César A.; Otárola, Enrique

doi:10.1016/j.cma.2019.112703

Cited by 16 publications

(5 citation statements)

References 29 publications

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“…Notice that q ′ ≥ d. Choosing q = d/(d−1) and utilizing a standard Sobolev embedding yield estimate (3). We now prove inequality (4).…”

Section: Weighted Function Spaces and Their Embeddingsmentioning

confidence: 82%

“…The purpose of this work is to study existence, uniqueness, and approximation results for the so-called Boussinesq model of thermally driven convection. While this problem has been considered before in different contexts and there are such results already available in the literature [3,9,12,18,29,36,37,38], our main source of novelty and originality here is that we allow the heat source to be singular, say a Dirac measure concentrated in a lower dimensional object so that the problem cannot be understood with the usual energy setting; as it was done, for instance, in [9,18,29,37,38]. Let us make this discussion precise.…”

mentioning

confidence: 99%

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The stationary Boussinesq problem under singular forcing

Allendes¹,

Otárola²,

Salgado³

2020

Preprint

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In Lipschitz two and three dimensional domains, we study the existence for the socalled Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to H −1 (̟, Ω), where ̟ is a weight in the Muckenhoupt class A 2 that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and ̟ −1 ∈ A 1 , show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability and local efficiency.

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“…Notice that q ′ ≥ d. Choosing q = d/(d−1) and utilizing a standard Sobolev embedding yield estimate (3). We now prove inequality (4).…”

Section: Weighted Function Spaces and Their Embeddingsmentioning

confidence: 82%

mentioning

confidence: 99%

The stationary Boussinesq problem under singular forcing

Allendes¹,

Otárola²,

Salgado³

2020

Preprint

Self Cite

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“…The approximation of errors in elliptical partial differential equations, such as those which govern linear elastic stress analysis, has received a great deal of attention 3‐6 . In many cases, error measures are utilized to drive mesh refinement through, for example, polytree decomposition algorithms 4,7‐10 . Error estimates have been developed for a wide range of solution approximation methods.…”

Section: Introductionmentioning

confidence: 99%

“…[3][4][5][6] In many cases, error measures are utilized to drive mesh refinement through, for example, polytree decomposition algorithms. 4,[7][8][9][10] Error estimates have been developed for a wide range of solution approximation methods. In addition to "conventional" FE, error estimates have been derived for boundary element methods, 8 immersed surface methods, 11 multigrid and composite FE methods, 7,12,13 extended FE (XFEM), 1,14 and stress singularity problems.…”

Section: Introductionmentioning

confidence: 99%

A probabilistic data assimilation framework to reconstruct finite element error fields from sparse error estimates: Application to sub‐modeling

Rouse

Kerfriden

Hamadi

2022

Numerical Meth Engineering

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The present work proposes a computational approach that recovers full finite element error fields from a small number of estimates of errors in scalar quantities of interest. The approach is weakly intrusive and is motivated by large scale industrial applications wherein modifying the finite element models is undesirable and multiple regions of interest may exist in a single model. Error estimates are developed using a Zhu-Zienkiewicz estimator coupled with the adjoint methodology to deliver goal-oriented results. A Bayesian probabilistic estimation framework is deployed for full field estimation. An adaptive, radial basis function based reduced order modeling strategy is implemented to reduce the cost of calculating the posterior. The Bayesian reconstruction approach, accelerated by the proposed model reduction technology, is shown to yield good probabilistic estimates of full error fields, with a computational complexity that is acceptable compared to the evaluation of the goal-oriented error estimates. The novelty of the work is that a set of computed error estimates are considered as partial observations of an underlying error field, which is to be recovered. Future improvements of the method include the optimal selection of goal-oriented error measures to be acquired prior to the error field reconstruction.

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“…For instance, in the context of phase change models there are some results based on error-related metric change [21,34] and on goal-oriented adaptivity [41]. Regarding the design and rigorous analysis of residual-based a posteriori error estimators for flow-transport couplings, the literature is predominantly focused on the stationary case (see, e.g., [3,6,7,9,13,22,37,42] and the references therein). Only a few results are available for the time-dependent regime, from which we mention the adaptive mixed method for Richards equation in porous media [15], the remeshing scheme based on goal-oriented adaptivity for solidification problems advanced in [14], the collection of adaptive schemes for reactive flow discussed in [16] and for heat transfer in [30].…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

Divergence-conforming methods for transient doubly-diffusive flows: A priori and a posteriori error analysis

Bürger,

Khan,

Méndez

et al. 2021

Preprint

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The analysis of the double-diffusion model and H(div)-conforming method introduced in [Bürger, Méndez, Ruiz-Baier, SINUM (2019), 57:1318-1343 is extended to the time-dependent case. In addition, the efficiency and reliability analysis of residual-based a posteriori error estimators for the steady, semi-discrete, and fully discrete problems is established. The resulting methods are applied to simulate the sedimentation of small particles in salinity-driven flows. The method consists of Brezzi-Douglas-Marini approximations for velocity and compatible piecewise discontinuous pressures, whereas Lagrangian elements are used for concentration and salinity distribution. Numerical tests confirm the properties of the proposed family of schemes and of the adaptive strategy guided by the a posteriori error indicators.

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Stabilized finite element approximations for a generalized Boussinesq problem: A posteriori error analysis

Cited by 16 publications

References 29 publications

The stationary Boussinesq problem under singular forcing

The stationary Boussinesq problem under singular forcing

A probabilistic data assimilation framework to reconstruct finite element error fields from sparse error estimates: Application to sub‐modeling

Divergence-conforming methods for transient doubly-diffusive flows: A priori and a posteriori error analysis

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