Abstract
In this work we introduce and analyze a mixed virtual element method for the two-dimensional nonlinear Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. For the continuous formulation we consider a dual-mixed approach in which the main unknowns are given by the gradient of the velocity and the pseudostress, whereas the velocity itself and the pressure are computed via simple postprocessing formulae. In addition, because of analysis reasons we add a redundant term arising from the constitutive equation relating the pseudostress and the velocity, so that the well-posedness of the resulting augmented formulation is established by using known results from nonlinear functional analysis. Then, we introduce the main features of the mixed virtual element method, which employs an explicit piecewise polynomial subspace and a virtual element subspace for approximating the aforementioned main unknowns, respectively. In turn, the associated computable discrete nonlinear operator is defined in terms of the \(\mathbb {L}^2\)-orthogonal projector onto a suitable space of polynomials, which allows the explicit integration of the terms involving deviatoric tensors that appear in the original setting. Next, we show the well-posedness of the discrete scheme and derive the associated a priori error estimates for the virtual element solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken \(\mathbb {H}(\mathbf {div})\)-norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.
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This work was partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, and the Becas-CONICYT Programme for foreign students; by Centro de Investigación en Ingeniería Matemática (CI\(^2\)MA), Universidad de Concepción; and by Universidad Nacional (Costa Rica), through the Project 0106-16.
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Gatica, G.N., Munar, M. & Sequeira, F.A. A mixed virtual element method for a nonlinear Brinkman model of porous media flow. Calcolo 55, 21 (2018). https://doi.org/10.1007/s10092-018-0262-7
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DOI: https://doi.org/10.1007/s10092-018-0262-7
Keywords
- Nonlinear Brinkman model
- Augmented formulation
- Virtual element method
- A priori error analysis
- Postprocessing techniques
- High-order approximations