A mixed finite element method for Darcy’s equations with pressure dependent porosity

by · Sep 25, 2014 · 1 327 views ·

Giordano Tierra, Charles University in Prague, Faculty of Mathematics and Physics, Prague, Czech Republic Joint work with Gabriel N. Gatica and Ricardo Ruiz-Baier In this talk I will present the a priori and a posteriori error analyses of a mixed finite element method for Darcy’s equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuška-Brezzi theory and the Banach fixed point Theorem. Next, we derive a reliable and efficient residual-based a posteriori error estimator for this problem. Finally, several numerical results illustrating the good performance of the method, confirming the aforementioned properties of the estimator, and showing the behavior of the associated adaptive algorithm, will be reported.

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