A Posteriori Error Analysis of a Mixed Finite Element Method for the Coupled Brinkman–Forchheimer and Double-Diffusion Equations

Caucao, Sergio; Gatica, Gabriel N.; Oyarzúa, Ricardo; Zúñiga, Paulo

doi:10.1007/s10915-022-02010-7

Cited by 4 publications

(1 citation statement)

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“…In this section we follow to [3, 4, 13, 14], and [15], and introduce an alternative a posteriori error estimator for the scheme (3.21a) and (3.21b) which is obtained after performing a different algebraic manipulation of the term

\left\Vert {\mathcal{R}}^{\mathtt{Flow}}\right\Vert

in (4.29) (cf. Lemma 4.4) allowing us to prove the respective reliability property without owing to a Helmholtz decomposition, in contrast to the previous one

\boldsymbol{\Theta}

.…”

Section: Residual‐based a Posteriori Error Estimator: The 2d Casementioning

confidence: 99%

A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media

Álvarez,

Colmenares,

Sequeira

2024

Numerical Methods Partial

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This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.

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\left\Vert {\mathcal{R}}^{\mathtt{Flow}}\right\Vert

in (4.29) (cf. Lemma 4.4) allowing us to prove the respective reliability property without owing to a Helmholtz decomposition, in contrast to the previous one