Publication Date:
2016-04-05
Description:
We propose and analyse a mixed finite element method for the nonstandard pseudostress–velocity formulation of the Stokes problem with varying density $\rho$ in $\mathbb {R}^d$ , $d\in \{2,3\}$ . Since the resulting variational formulation does not have the standard dual-mixed structure, we reformulate the continuous problem as an equivalent fixed-point problem. Then, we apply the classical Babuška–Brezzi theory to prove that the associated mapping $\mathbb {T}$ is well defined, and assuming that $\|{\nabla \rho }/{\rho }\|_{{\textbf{L}}^{\infty }(\Omega )}$ is sufficiently small, we show that $\mathbb {T}$ is a contraction mapping, which implies that the variational formulation is well posed. Under the same hypothesis on $\rho$ we prove stability of the continuous problem. Next, adapting the arguments of the continuous analysis to the discrete case, we establish suitable hypotheses on the finite element subspaces ensuring that the associated Galerkin scheme becomes well posed. A feasible choice of subspaces is given by Raviart–Thomas elements of order $k \geq 0$ for the pseudostress and polynomials of degree $k$ for the velocity. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the problem. The proof of reliability makes use of the global inf–sup condition, Helmholtz decompositions, and local approximation properties of the Clément interpolant and Raviart–Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, approximation properties of the $L^2$ -orthogonal projector and known results from previous works are the main tools for proving the efficiency of the estimator. Finally, several numerical results illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms are reported.
Print ISSN:
0272-4979
Electronic ISSN:
1464-3642
Topics:
Mathematics
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