A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem
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Álvarez Guadamuz, Mario Andrés
Gatica Pérez, Gabriel Nibaldo
Ruiz Baier, Ricardo
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Abstract
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 fullymixed 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.
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Keywords
Brinkman–Darcy equations, Vorticity-based formulation, Mixed finite element methods, A posteriori error analysis, MATEMÁTICAS
Citation
https://link.springer.com/article/10.1007/s10092-017-0238-z