An augmented velocity–vorticity–pressure formulation for the Brinkman equations

Abstract

SUMMARYThis paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity and pressure, for the Brinkman problem with non-standard boundary conditions. The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive equation relating the aforementioned unknowns, and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic which, thanks to the Lax-Milgram Theorem, and besides proving the well-posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart-Thomas elements of any order k ≥ 0 for the velocity field, and piecewise continuous polynomials of degree k + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual-based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators, and showing the behaviour of the associated adaptive algorithms, are reported. Copyright c 2014 John Wiley & Sons, Ltd.Received . . . KEY WORDS: Generalized Stokes equations; Brinkman problem; vorticity-based mixed formulation; augmented scheme; finite element method; a priori error estimates; a posteriori error analysis.

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

“…Estimate (b) can be derived by adapting the arguments in the proof of [2,Lemma 4.15]. Finally, since div(ω B ) = div(curl u B ) = 0 in B , for the derivation of (c), it suffices to apply (3.30) to ζ := ω B and ζ h := ω B h .…”

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