Ricardo G. Durán

Mixed finite elements for second order elliptic problems in three variables

SummaryTwo families of mixed finite elements, one based on simplices and the other on cubes, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. These spaces are analogues of those introduced by Brezzi, Douglas, and Marini in two space variables. Error estimates inL2 andH−s are derived.

On mixed finite element methods for the Reissner-Mindlin plate model

In this paper we analyze the convergence of mixed finite element approximations to the solution of the Reissner-Mindlin plate problem. We show that several known elements fall into our analysis, thus providing a unified approach. We also introduce a low-order triangular element which is optimalorder convergent uniformly in the plate thickness.

A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems

This paper deals with a posteriori error estimators for the linear finite element approximation of second-order elliptic eigenvalue problems in two or three dimensions. First, we give a simple proof of the equivalence, up to higher order terms, between the error and a residual type error estimator. Second, we prove that the volumetric part of the residual is dominated by a constant times the edge or face residuals, again up to higher order terms. This result was not known for eigenvalue problems.

Mixed Finite Elements, Compatibility Conditions, and Applications

Since the early 70’s, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and their use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay between theory and application. Discretization spaces for mixed schemes require suitable compatibilities, so that simple minded approximations generally do not work and the design of appropriate stabilizations gives rise to challenging mathematical problems.

Finite element vibration analysis of fluid-solid systems without spurious modes

This paper deals with the finite element approximation of the vibration modes of a problem with fluid–structure interaction. Displacement variables are used for both the fluid and the solid. To avoid the typical spurious modes of this formulation we introduce a nonconforming discretization. Error estimates for the approximation of eigenvalues and eigenvectors are given.

Error estimators for nonconforming finite element approximations of the Stokes problem

In this paper we define and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations. We prove that these estimators are equivalent to an appropriate norm of the error. For the case of piecewise linear elements we define two estimators. Both of them are easy to compute, but the second is simpler because it can be computed using only the right-hand side and the approximate velocity. We show how the first estimator can be generalized to higher-order elements. Finally, we present several numerical examples in which one of our estimators is used for adaptive refinement.

Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements

This paper adresses the problem of determining upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method. These bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes. They depend strongly on the geometry of the triangles and (relatively weakly) on the smoothness of the solution. An example shows that the bounds are not over pessimistic. In Babuska, Plank, and Rodriguez (4) detailed numerical experimentation is given.

The Maximum Angle Condition for Mixed and Nonconforming Elements: Application to the Stokes Equations

For the Lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition. The objective of this paper is to obtain similar results for the Raviart--Thomas interpolation arising in the analysis of mixed methods. We prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and, moreover, that this condition is indeed necessary to have these estimates. Error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the Stokes equations are derived from our results.

On the asymptotic exactness of error estimators for linear triangular finite elements

SummaryThis paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.

Error estimates for moving least square approximations

Abstract In this paper we obtain error estimates for moving least square approximations in the one-dimensional case. For the application of this method to the numerical solution of differential equations it is fundamental to have error estimates for the approximations of derivatives. We prove that, under appropriate hypothesis on the weight function and the distribution of points, the method produces optimal order approximations of the function and its first and second derivatives. As a consequence, we obtain optimal order error estimates for Galerkin approximations of coercive problems. Finally, as an application of the moving least square method we consider a convection–diffusion equation and propose a way of introducing up-wind by means of a non-symmetric weight function. We present several numerical results showing the good behavior of the method.