Dr. Gatica-Simpertigue, Luis

2016, Dr. Gatica-Simpertigue, Luis, Gatica, Gabriel, Sequeira, Filánder

In this paper we present the apriori and aposteriori error analyses of a non-standard mixed finite element method for the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. More precisely, the approach introduced here is based on a simplified interpretation of the pseudostress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. In addition, physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through a simple postprocessing in terms of the pseudostress variable. Furthermore, we also introduce a second elementby-element postprocessing formula for the stress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. We apply the classical Babuška–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of order k ≥ 0 for the pseudostress and piece wise polynomials of degree≤ k for the displacement can be utilized. Moreover, were mark that in the 3D case the number of unknowns behaves approximately as 9 times the number of elements (tetrahedra) of the triangulation when k = 0. This factor increases to 12.5 when one uses the classical PEERS. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the mixed finite element scheme. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the associated adaptive algorithm, are provided.

2015, Dr. Gatica-Simpertigue, Luis, Gatica, Gabriel, Sequeira, Filánder

We present a non-standard mixed finite element method for the linear elasticity problem in Rn with non-homogeneous Dirichlet boundary conditions. More precisely, our approach his based on a simplified interpretation of the pseudo stress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. We apply the classical Babuˇ ska–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of orderk≥0 for the pseudo stress and piece wise polynomials of degree ≤k for the displacement can be utilized. In addition, complementing the results in the afore mentioned reference, we introduceanewpostprocessingformulaforthestressrecoveringtheoptimally convergent approximation of the broken H(div)-norm. Numerical results confirm our theoretical findings.

2015, Gatica, Gabriel N., Gatica-Simpertigue, Luis, Sequeira, Filander A.

In this paper we introduce and analyze an augmented mixed finite element method for the twodimensional nonlinear Brinkman model of porous media flow with mixed boundary conditions. More precisely, we extend a previous approach for the respective linear model to the present nonlinear case, and employ a dual-mixed formulation in which the main unknowns are given by the gradient of the velocity and the pseudostress. In this way, and similarly as before, the original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, a feasible choice of finite element subspaces is given by Raviart-Thomas elements of order k ≥ 0 for the pseudostress, piecewise polynomials of degree ≤ k for the gradient, and continuous piecewise polynomials of degree ≤ k + 1 for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are provided.

2018, Gatica-Simpertigue, Luis, Oyarzúa, Ricardo, Sánchez, Nestor

We introduce and analyze an augmented mixed finite element method for the Navier–Stokes–Brinkman problem with nonsolenoidal velocity. We employ a technique previously applied to the stationary Navier–Stokes equation, which consists of the introduction of a modified pseudostress tensor relating the gradient of the velocity and the pressure with the convective term, and propose an augmented pseudostress–velocity formulation for the model problem. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Banach fixed point theorem, combined with the Lax–Milgram lemma, are applied to prove the unique solvability of the continuous and discrete systems. We point out that no discrete inf–sup conditions are required for the solvability analysis, and hence, in particular for the Galerkin scheme, arbitrary finite element subspaces of the respective continuous spaces can be utilized. For instance, given an integer k≥0, the Raviart–Thomas spaces of order k and continuous piecewise polynomials of degree ≤k+1 constitute feasible choices of discrete spaces for the pseudostress and the velocity, respectively, yielding optimal convergence. We also emphasize that, since the Dirichlet boundary condition becomes a natural condition, the analysis for both the continuous an discrete problems can be derived without introducing any lifting of the velocity boundary datum. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed method. The proof of reliability makes use of a global inf–sup condition, a Helmholtz decomposition, 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 L2-orthogonal projector, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results illustrating the performance of the augmented mixed method, confirming the theoretical rate of convergence and properties of the estimator, and showing the behavior of the associated adaptive algorithms, are reported.