<p>We introduce and analyze new Banach spaces-based fully-mixed finite element methods for the convective Brinkman-Forchheimer equations coupled with a nonlinear transport phenomenon. Our approach is based on the incorporation of the fluid velocity gradient, the incomplete nonlinear fluid pseudostress, the concentration gradient, and the total (diffusive plus advective) flux for the concentration, as auxiliary variables, which, along with the velocity and concentration themselves, constitute the set of unknowns of the model. The resulting mixed variational formulation can be written as two coupled nonlinear saddle point systems, which are then reformulated as an equivalent fixed-point equation defined in terms of the operators solving the corresponding decoupled problems. An analogue approach is utilized for the associated Galerkin scheme. In this way, the Babuška-Brezzi theory, some abstract results on monotone operators, and the classical Banach fixed-point theorem are employed to establish the well-posedness of both the continuous and discrete schemes. In particular, for each integer <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2958_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$k \,\ge \, 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mspace width="0.166667em" /> <mo>≥</mo> <mspace width="0.166667em" /> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, vector and tensor Raviart-Thomas subspaces of order <i>k</i> for the pseudostress and the total flux, respectively, as well as piecewise polynomial subspaces of degree <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2958_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\le k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> for the velocity, the concentration, and their respective gradients, yield stable Galerkin schemes. Optimal a priori error estimates along with the corresponding rates of convergence are also established. Finally, several numerical experiments confirming the latter and illustrating the good performance of the method in 2D and 3D, are reported.</p>

New Fully Mixed Finite Element Methods for the Coupled Convective Brinkman-Forchheimer and Nonlinear Transport Equations

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发表日期：

2025年7月2日

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Alonso J. Bustos,
Sergio Caucao,
Gabriel N. Gatica,
Benjamín N. Venegas

摘要

We introduce and analyze new Banach spaces-based fully-mixed finite element methods for the convective Brinkman-Forchheimer equations coupled with a nonlinear transport phenomenon. Our approach is based on the incorporation of the fluid velocity gradient, the incomplete nonlinear fluid pseudostress, the concentration gradient, and the total (diffusive plus advective) flux for the concentration, as auxiliary variables, which, along with the velocity and concentration themselves, constitute the set of unknowns of the model. The resulting mixed variational formulation can be written as two coupled nonlinear saddle point systems, which are then reformulated as an equivalent fixed-point equation defined in terms of the operators solving the corresponding decoupled problems. An analogue approach is utilized for the associated Galerkin scheme. In this way, the Babuška-Brezzi theory, some abstract results on monotone operators, and the classical Banach fixed-point theorem are employed to establish the well-posedness of both the continuous and discrete schemes. In particular, for each integer $k \,\ge \, 0$ $k \geq 0$ , vector and tensor Raviart-Thomas subspaces of order k for the pseudostress and the total flux, respectively, as well as piecewise polynomial subspaces of degree $\le k$ $\leq k$ for the velocity, the concentration, and their respective gradients, yield stable Galerkin schemes. Optimal a priori error estimates along with the corresponding rates of convergence are also established. Finally, several numerical experiments confirming the latter and illustrating the good performance of the method in 2D and 3D, are reported.