Mixed Finite Element Methods (original) (raw)

A Unified Theory For The Numerical Analysis Of Mixed And Stabilized Finite Element Methods

This paper reports the mathematical analysis of the Augmented Mixed (AM) Method to solve the equations of steady incompressible flows. AM Method is an internal approximation of an approximated variational formulation of the equations. This method includes as particular cases Mixed and Stabilized Finite Element Methods. The main feature of AM Method is that it bounds a H 1 norm of the high-frequences component of the residual that are representable in a specific space, by means of a static condensation operator. We consider both the Generalized Stokes equations (including an advection term) and the Navier-Stokes equations.

A discourse on the stability conditions for mixed finite element formulations

Computer Methods in Applied Mechanics and Engineering, 1990

We discuss the general mathematical conditions for solvability, stability and optimal error bounds of mixed finite element discretizations. Our objective is to present these conditions with relatively simple arguments. We present the conditions for solvability and stability by considering the general coefficient matrix of mixed finite element discretizations, and then deduce the conditions for optimal error bounds for the distance between the finite element solutions and the exact solution of the mathematical problem. To exemplify our presentation we consider the solutions of various example problems. Finally, we also present a numerical test that is useful to identify numerically whether, for the solution of the general Stokes flow problem, a given finite element discretization satisfies the stability and optimal error bound conditions.

A Simple Introduction to the Mixed Finite Element Method

SpringerBriefs in Mathematics, 2014

SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians.

Mixed stabilized finite element methods in nonlinear solid mechanics Part I: Formulation

This paper exploits the concept of stabilized finite element methods to formulate stable mixed stress/ displacement and strain/displacement finite elements for the solution of nonlinear solid mechanics problems. The different assumptions and approximations used to derive the methods are exposed. The proposed procedure is very general, applicable to 2D and 3D problems. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the results obtained compare favorably with those obtained with the corresponding irreducible formulation.

Mixed finite element methods for elliptic problems

Computer Methods in Applied Mechanics and Engineering, 1990

This paper treats the basic ideas of mixed finite element methods at an introductory level. Although the viewpoint presented is that of a mathematician, the paper is aimed at practitioners and the mathematical prerequisites are kept to a minimum. A classification of variational principles and of the corresponding weak formulations and Galerkin methods-displacement, equilibrium, and mixed-is given and illustrated through four significant examples. The advantages and disadvantages of mixed methods are discussed. The concepts of convergence, approximability, and stability and their interrelations are developed, and a résumé is given of the stability theory which governs the performance of mixed methods. The paper concludes with a survey of techniques that have been developed for the construction of stable mixed methods and numerous examples of such methods.

Dual formulations of mixed finite element methods with applications

Computer-Aided Design, 2011

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail. k ) −1 . We show how the choice of discrete Hodge star requires certain geometric quality conditions of the primal and dual mesh elements. A specific example is given showing how our dual formulation of the problem can result in a better conditioned linear system than the primal formulations.

A generalization for stable mixed finite elements

Proceedings of the 14th ACM Symposium on Solid and Physical Modeling - SPM '10, 2010

Mixed finite element methods solve a PDE involving two or more variables. In typical problems from electromagnetics and electrodiffusion, the degrees of freedom associated to the different variables are stored on both primal and dual domain meshes and a discrete Hodge star is used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the model and numerical stability of a finite element method. We also show how to define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods.

An analysis of the convergence of mixed finite element methods

RAIRO. Analyse numérique

An analysis of the convergence of mixed finite element methods RAIRO-Analyse numérique, tome 11, n o 4 (1977), p. 341-354. http://www.numdam.org/item?id=M2AN\_1977\_\_11\_4\_341\_0 © AFCET, 1977, tous droits réservés. L'accès aux archives de la revue « RAIRO-Analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/ legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Dual Formulations of Mixed Finite Element Methods

arXiv (Cornell University), 2010

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.

Mixed methods using standard conforming finite elements

Computer Methods in Applied Mechanics and …, 2009

We investigate the mixed finite element method (MFEM) for solving a second order elliptic problem with a lowest order term, as might arise in the simulation of single phase flow in porous media. We find that traditional mixed finite element spaces are not necessary when a positive lowest order (i.e., reaction) term is present. Hence we propose to use standard conforming finite elements Q k × (Q k ) d on rectangles or P k × (P k ) d on simplices to solve for both the pressure and velocity field in d dimensions. The price we pay is that we have only sub-optimal order error estimates. With a delicate superconvergence analysis, we find some improvement for the simplest pair Q k × (Q k ) d with any k ≥ 1, or for P 1 × (P 1 ) d , when the mesh is uniform and the solution has one extra order of regularity. We also prove similar results for both parabolic and second order hyperbolic problems. Numerical results using Q 1 × (Q 1 ) 2 and P 1 × (P 1 ) 2 are presented in support of our analysis. These observations allow us to simplify the implementation of the MFEM, especially for higher order approximations, as might arise in an hp-adaptive procedure. : 65N30.