Triangular Mixed Finite Element Research Articles

On the construction of a triangular Mixed Finite Element based on the principle of Hellinger‐Reissner

AbstractA low order finite element based on a Hellinger‐Reissner formulation with weak enforcement of the stress symmetry is presented. The novel element gets along without any additional degrees of freedom beside the displacements and stresses. Therein, lowest order Raviart‐Thomas interpolation is utilized for the stresses and a C0 continuous polynomial interpolation of first order is adopted for the displacements. This combination leads to a very efficient and robust element formulation which will be depicted on a couple of numerical examples.

Superconvergence of triangular mixed finite element methods for nonlinear optimal control problems

Superconvergence of triangular mixed finite element methods for nonlinear optimal control problems

Optimization techniques for mixed finite element methods of an semilinear elliptic PDEs - superconvergence

In this paper, we shall investigate the superconvergence property of semilinear quadratic elliptic optimal control problems by triangular mixed finite element methods. The state and co-state are approximated by Raviart-Thomas mixed finite element spaces of order k, and the control is approximated by piecewise constant functions. We derive the superconvergence properties for both the control variable and the state variables.

Error estimates of triangular mixed finite element methods for quasilinear optimal control problems

The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.

Higher Order Triangular Mixed Finite Element Methods for Semilinear Quadratic Optimal Control Problems

In this paper, we investigate a priori error estimates for the quadratic opti- mal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods. The state and the co-state are ap- proximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k (k≥ 0). A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained. Finally, we present some numerical examples which confirm our theoretical results. AMS subject classifications: 49J20, 65N30

Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint

In this paper, we shall investigate the superconvergence property of quadratic elliptical optimal control problems by triangular mixed finite element methods. The state and co-state are approximated by the order k = 1 Raviart–Thomas mixed finite elements and the control is discretized by piecewise constant functions. We prove the superconvergence error estimate of h 2 in L 2-norm between the approximated solution and the interpolation of the exact control variable. Moreover, by postprocessing technique, we find that the projection of the discrete adjoint state is superclose (in order h 2) to the exact control variable.

L ∞-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations

We look at L ∞-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations. In so doing, use is made of mixed finite element methods. The state and costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control, by piecewise constant functions. L ∞-error estimates of optimal order are derived for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, numerical tests are presented which confirm our theoretical results.

Superconvergence of quadratic optimal control problems by triangular mixed finite element methods

AbstractThe aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method. The state and co‐state are approximated by the order k⩽1 Raviart–Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. We construct an interpolation of the exact control and a projection of the discrete scalar co‐state to be the approximated solution of the control variable for the continuous optimal control problem. As a result, it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h3/2 for k=0 and of order h2 for k=1. Moreover, only for the order k=1 Raviart–Thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h2. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence. Copyright © 2008 John Wiley & Sons, Ltd.

A New Formulation of the Mixed Finite Element Method for Solving Elliptic and Parabolic PDE with Triangular Elements

For the groundwater flow problem (which corresponds to the Darcy flow model), we show how to produce a scheme with one unknown per element, starting from a mixed formulation discretized with the Raviart Thomas triangular elements of lowest order. The aim is here to obtain a new formulation with one unknown per element by elimination of the velocity variables q=−k grad P, without any restriction concerning the computation of the velocity field. In the first part, we describe the triangular mixed finite element method used for solving Darcy's and mass balance equations. In the second part, we study the elliptic–parabolic problem; we describe the new formulation of the problem in order to use mixed finite elements (MFE) with less unknowns without any specific numerical integration. Finally, we compare the computational effort of the MFE method with the new formulation for different triangulations using numerical experiments. In this work, we show that the new formulation can be seen as a general formulation which can be equivalent to the finite volume or the finite difference methods in some particular cases.

Analysis of a mixed finite element method for the Reissner-Mindlin plate problems

An analysis of a triangular mixed finite element method, proposed by Taylor, and Auricchio (cf. [13]) is presented. The method is based on a linked interpolation between deflections, and rotations in order to avoid the locking phenomenon (cf. [15]). The analysis shows that the approximated deflections, and rotations are first order convergent to the exact solution, uniformly in the thickness.

Superconvergence and a posteriori error estimation for triangular mixed finite elements

In this paper,we prove superconvergence results for the vector variable when lowest order triangular mixed finite elements of Raviart-Thomas type [17] on uniform triangulations are used, i.e., that the $H(\mbox{div;} \Omega)$ -distance between the approximate solution and a suitable projection of the real solution is of higher order than the $H(\mbox{div;} \Omega)$ -error. We prove results for both Dirichlet and Neumann boundary conditions. Recently, Duran [9] proved similar results for rectangular mixed finite elements, and superconvergence along the Gauss-lines for rectangular mixed finite elements was considered by Douglas, Ewing, Lazarov and Wang in [11], [8] and [18]. The triangular case however needs some extra effort. Using the superconvergence results, a simple postprocessing of the approximate solution will give an asymptotically exact a posteriori error estimator for the $L^2(\Omega)$ -error in the approximation of the vector variable.

A triangular mixed finite element method for the stationary semiconductor device equations

On presente ici une methode d'elements finis mixte, de type Petrov-Galerkin, basee sur des elements triangulaires, pour un systeme elliptique auto-adjoint du second ordre, emanant d'un modele stationnaire pour des semiconducteurs. Cette methode est basee sur une nouvelle formulation du probleme discret correspondant et peut etre consideree comme une extension bidimensionnelle naturelle de la methode bien connue de Scharfetter-Gummel. L'existence, l'unicite et la stabilite de la solution approchee sont etablies pour un maillage triangulaire arbitraire et une estimation de l'erreur est donnee pour une triangulation de Delaunay arbitraire et sa tesselation de Dirichlet

Linear and non-linear analysis of thin shallow shells by mixed finite elements

Two triangular mixed finite elements are developed for the linear and non-linear analysis of thin shallow shells. The unknown variables at the finite element nodes include not only the three displacements and bending stress resultants, but also the membrane stress resultants. The theory of thin shallow shells based on Kirchhoff-Love's hypothesis, and Reissner's principle have been used to obtain explicitly the finite element matrices. The interpolation functions used for the nodal variables are linear for the first finite element and quadratic for the second one. The non-linear problem is solved by the method of Newton-Raphson and the results are compared to those obtained by exact and approximate methods.