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

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Abstract

In this paper we consider a partially augmented fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman–Forchheimer and double-diffusion equations, and develop an a posteriori error analysis for the 2D and 3D versions of the associated mixed finite element scheme. Indeed, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity and inf-sup condition of the forms involved, a suitable assumption on the data, stable Helmholtz decompositions in Hilbert and Banach frameworks, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, 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 examples confirming the theoretical properties of the estimators and illustrating the performance of the associated adaptive algorithms, are reported. In particular, the case of flow through a 3D porous media with channel networks is considered.

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Authors and Affiliations

  1. Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

    Sergio Caucao

  2. CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Gabriel N. Gatica & Ricardo Oyarzúa

  3. GIMNAP-Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

    Ricardo Oyarzúa

  4. Department of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Paulo Zúñiga

Authors
  1. Sergio Caucao
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  2. Gabriel N. Gatica
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  3. Ricardo Oyarzúa
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  4. Paulo Zúñiga
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Correspondence to Sergio Caucao.

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This research was partially supported by ANID-Chile through the projects, Centro de Modelamiento Matemático (FB210005), Anillo of Computational Mathematics for Desalination Processes (ACT210087), and Fondecyt projects 1200666 and 11220393; by Centro de Investigación en Ingeniería Matemática (\(\hbox {CI}^2\)MA), Universidad de Concepción; and by Universidad del Bío-Bío through VRIP-UBB project 2120173 GI/C.

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Caucao, S., Gatica, G.N., Oyarzúa, R. et al. A Posteriori Error Analysis of a Mixed Finite Element Method for the Coupled Brinkman–Forchheimer and Double-Diffusion Equations. J Sci Comput 93, 50 (2022). https://doi.org/10.1007/s10915-022-02010-7

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