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

Journal article

We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fully-mixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68–95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart–Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.

Authors: Alvarez, M; Gatica, G; Ruiz Baier, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer Verlag
• Journal: Calcolo
• Volume: 54
• Issue: 4
• Pages: 1491-1519
• Publication date: 2017-09-05
• Acceptance date: 2017-08-29
• DOI: 10.1007/s10092-017-0238-z
• EISSN: 1126-5434
• ISSN: 0008-0624
• Keywords: Mixed finite element methods; Vorticity-based formulation; A posteriori error analysis; Brinkman–Darcy equations