A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model

Colmenares, Eligio; Gatica, Gabriel N.; Oyarzúa, Ricardo

doi:10.1016/j.camwa.2018.10.009

Cited by 12 publications

(11 citation statements)

References 23 publications

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“…and proceeding analogously to Lemma 3.9 (see also [6], Lemma 3.7), we obtain the remaining terms of 3.36. Further details are omitted.…”

Section: Lemma 311mentioning

confidence: 80%

“…In particular, using the alternative definition of ℛ h (cf., 3.24) and proceeding similarly to Lemma 3.8, we find the first, second, and last term of the local estimator 3.36. On the other hand, considering the original definition 3.16 of ℛ h , noting that

\frac{d θ_{normalD}}{d s} \in {normalL}^{2} (Γ_{D})

, applying the integration by parts formula on Γ D given by (cf., [ , Lemma 3.5, equation 3.34])

{(⟨⟩,, normalcurl χ \cdot boldn, θ_{normalD})}_{Γ_{D}} = - {(⟨⟩,, \frac{d θ_{normalD}}{d s}, χ)}_{Γ_{D}} \forall χ \in {normalH}^{1} (Ω)

and proceeding analogously to Lemma 3.9 (see also , Lemma 3.7), we obtain the remaining terms of 3.36. Further details are omitted.…”

Section: A Posteriori Error Analysis: the 2‐d Casementioning

confidence: 84%

“…Proof It follows almost straightforwardly from a slight modification of the proof of Ref. (Lemma 3.11). We omit further details.…”

Section: A Posteriori Error Analysis: the 2‐d Casementioning

confidence: 95%

“…According to the above discussion and to complement the study, started in Ref. [1] for the nonisothermal Oldroyd-Stokes problem, in this article, we proceed similarly to [5,6,13,14], and develop two reliable and efficient residual-based a posteriori error estimators for the augmented-mixed finite element method studied in Ref. [1].…”

Section: Introductionmentioning

confidence: 99%

“…For example, residual‐based a posteriori error analyses for the aforementioned Boussinesq model have been developed in Refs. and for the associated mixed‐primal and fully mixed formulations, respectively. In fact, standard arguments relying on duality techniques, suitable decompositions and classical approximation properties are combined there with corresponding small data assumptions to derive the reliability of the estimators.…”

Section: Introductionmentioning

confidence: 99%

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A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

Caucao

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Oyarzúa

2018

Numerical Methods Partial

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In this article, we consider an augmented fully mixed variational formulation that has been recently proposed for the nonisothermal Oldroyd–Stokes problem, and develop an a posteriori error analysis for the 2‐D and 3‐D versions of the associated mixed finite element scheme. More precisely, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem on arbitrary (convex or nonconvex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the bilinear forms of the continuous formulation, suitable assumptions on the domain and the data, stable Helmholtz decompositions, and the local approximation properties of the Clément and Raviart–Thomas operators. On the other hand, inverse inequalities, the localization technique based on bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the a posteriori error estimators and illustrating the performance of the associated adaptive algorithms are reported.

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“…and proceeding analogously to Lemma 3.9 (see also [6], Lemma 3.7), we obtain the remaining terms of 3.36. Further details are omitted.…”

Section: Lemma 311mentioning

confidence: 80%

\frac{d θ_{normalD}}{d s} \in {normalL}^{2} (Γ_{D})

, applying the integration by parts formula on Γ D given by (cf., [ , Lemma 3.5, equation 3.34])

{(⟨⟩,, normalcurl χ \cdot boldn, θ_{normalD})}_{Γ_{D}} = - {(⟨⟩,, \frac{d θ_{normalD}}{d s}, χ)}_{Γ_{D}} \forall χ \in {normalH}^{1} (Ω)

and proceeding analogously to Lemma 3.9 (see also , Lemma 3.7), we obtain the remaining terms of 3.36. Further details are omitted.…”

Section: A Posteriori Error Analysis: the 2‐d Casementioning

confidence: 84%

“…Proof It follows almost straightforwardly from a slight modification of the proof of Ref. (Lemma 3.11). We omit further details.…”

Section: A Posteriori Error Analysis: the 2‐d Casementioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

Caucao

Gatica

Oyarzúa

2018

Numerical Methods Partial

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An Uzawa iterative method for the natural convection problem based on mixed finite element method

Huang

2021

Math Methods in App Sciences

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In this paper, an Uzawa iterative method combined with a mixed finite element method is presented for solving the natural convection model, where a decoupling discrete system is solved and no saddle point system is required to solve at each iterative step. Moreover, the authors establish convergence result of the method and find an interval of relaxation parameter that is crucial for the convergence of the presented method.

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Topology Optimization for Steady-State Anisothermal Flow Targeting Solids with Piecewise Constant Thermal Diffusivity

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A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model

Cited by 12 publications

References 23 publications

A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

An Uzawa iterative method for the natural convection problem based on mixed finite element method

Topology Optimization for Steady-State Anisothermal Flow Targeting Solids with Piecewise Constant Thermal Diffusivity

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