A Simple Introduction to the Mixed Finite Element Method

In this article, we propose and analyze a new mixed variational formulation for the stationary Boussinesq problem. Our method, which uses a technique previously applied to the Navier-Stokes equations, is based first on the introduction of a modified pseudostress tensor depending nonlinearly on the velocity through the respective convective term. Next, the pressure is eliminated, and an augmented approach for the fluid flow, which incorporates Galerkin-type terms arising from the constitutive and equilibrium equations, and from the Dirichlet boundary condition, is coupled with a primal-mixed scheme for the main equation modeling the temperature. In this way, the only unknowns of the resulting formulation are given by the aforementioned nonlinear pseudostress, the velocity, the temperature, and the normal derivative of the latter on the boundary. An equivalent fixed-point setting is then introduced and the corresponding classical Banach Theorem, combined with the Lax-Milgram Theorem and the Babuška-Brezzi theory, are applied to prove the unique solvability of the continuous problem. In turn, the Brouwer and the Banach fixed-point theorems are used to establish existence and uniqueness of solution, respectively, of the associated Galerkin scheme. In particular, Raviart-Thomas spaces of order k for the pseudostress, continuous piecewise polynomials of degree ≤ k+1 for the velocity and the temperature, and piecewise polynomials of degree ≤ k for the boundary unknown become feasible choices. Finally, we derive optimal a priori error estimates, and provide several numerical results illustrating the good performance of the augmented mixed-primal finite element method and confirming the theoretical rates of convergence.

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“…We end this section by recalling from [23], the approximation properties of the specific finite element subspaces introduced here.…”

Section: And the Same A Priori Estimates From Theorem 48 Holdmentioning

confidence: 99%

Analysis of an augmented mixed‐primal formulation for the stationary Boussinesq problem

Colmenares

Gatica

Oyarzúa

2015

Numerical Methods Partial

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“…This is a linear and continuous map (see [13,Propositions 3.4 and 3.6]) which will be required in the sequel. We will also make use of the Raviart-Thomas interpolator of lowest order (see [22]) h : H 1 ( ) → H h , ∈ {B, D}, which according to its characterisation given by the identity…”

Section: Preliminariesmentioning

confidence: 99%

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

Alvarez

Gatica

Ruiz–Baier

2017

Calcolo

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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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“…Further approximation properties of I h and Π h are summarised in the following lemmas (see a proof in e.g. [15] and [22,Lemmas 3.16 and 3.18], respectively).…”

Section: Preliminariesmentioning

confidence: 99%

A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

Gatica

Ruiz–Baier

Tierra

2016

Computers & Mathematics with Applications

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In this work we develop the a posteriori error analysis of an augmented mixed finite element method for the 2D and 3D versions of the Navier-Stokes equations when the viscosity depends nonlinearly on the module of the velocity gradient. Two different reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions are derived. Our analysis of reliability of the proposed estimators draws mainly upon the global inf-sup condition satisfied by a suitable linearization of the continuous formulation, an application of Helmholtz decomposition, and the local approximation properties of the RaviartThomas and Clément interpolation operators. In addition, differently from previous approaches for augmented mixed formulations, the boundedness of the Clément operator plays now an interesting role in the reliability estimate. On the other hand, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show their efficiency. Finally, several numerical results are provided to illustrate the good performance of the augmented mixed method, to confirm the aforementioned properties of the a posteriori error estimators, and to show the behaviour of the associated adaptive algorithm.

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A Simple Introduction to the Mixed Finite Element Method

Cited by 183 publications

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Analysis of an augmented mixed‐primal formulation for the stationary Boussinesq problem

Analysis of an augmented mixed‐primal formulation for the stationary Boussinesq problem

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

A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

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