Rolf Stenberg

A family of mixed finite elements for the elasticity problem

SummaryA new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both two-and three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.

On some techniques for approximating boundary conditions in the finite element method

We discuss the stabilization of finite element methods in which essential boundary conditions are approximated by Babuskas method of Lagrange multipliers and we show that there is a close connection with this technique and a classical method by Nitsche.

ERROR ANALYSIS OF MIXED-INTERPOLATED ELEMENTS FOR REISSNER-MINDLIN PLATES

We give an error analysis for the recently introduced mixed-interpolated finite element methods for Reissner-Mindlin plates. Optimal error estimates, which are valid uniformly with respect to the thickness of the plate, are proven for the deflection, rotation and the shear force. In addition, the earlier families are augmented with a new method with linear approximations for the deflection and the rotation. We also introduce a simple postprocessing method by which an improved approximation for the deflection can be obtained.

On the construction of optimal mixed finite element methods for the linear elasticity problem

SummaryThe mixed finite element method for the linear elasticity problem is considered. We propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement. The ideas are applied in some examples.

Error analysis of some finite element methods for the Stokes problem

We prove the optimal order of convergence for some two-dimensional finite element methods for the Stokes equations. First we consider methods of the Taylor-Hood type: the triangular P3 P2 element and the Qk Qk-1 I k > 2, family of quadrilateral elements. Then we introduce two new low-order methods with piecewise constant approximations for the pressure. The analysis is performed using our macroelement technique, which is reviewed in a slightly altered form.

A stable bilinear element for the Reissner-Mindlin plate model

Abstract In this paper, we introduce a new quadrilateral element using isoparametric bilinear basis functions for both components of the rotation vector and the deflection. The element is a stable modification of the MITC4 element. We report on calculations with this new element, the original MITC4 and also the bilinear element, with selective reduced integration. The numerical results are in accordance with the results of the numerical analysis and they show that (i) the method with reduced integration is highly unreliable and cannot be recommended; (ii) the MITC4 performs rather well, but its instability can lead to a decrease in accuracy for the deflection and especially to an inaccurate and oscillating shear force; (iii) the drawbacks of the MITC4 are not present in the modified method.

Nitsche’s method for general boundary conditions

We introduce a method for treating general boundary conditions in the finite element method generalizing an approach, due to Nitsche (1971), for approximating Dirichlet boundary conditions. We use Poissons equations as a model problem and prove a priori and a posteriori error estimates. The method is also compared with the traditional Galerkin method. The theoretical results are verified numerically.

A linear nonconforming finite element method for nearly incompressible elasticity and stokes flow

Abstract We introduce a new triangular element for nearly incompressible elasticity and incompressible fluid flow. The method consists of conforming linear elements for one of the displacement (or velocity for flows) component and linear non-conforming elements for the other component. The element is proved to give an optimal approximation and this is also confirmed by several numerical examples.

Energy norm a posteriori error estimates for mixed finite element methods

This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different ways: under a saturation assumption and using a Helmholtz decomposition for vector fields.

An optimal low-order locking-free finite element method for Reissner-Mindlin plates

We propose a simple modification of a recently introduced locking-free finite element method for the Reissner–Mindlin plate model. By this modification, we are able to obtain optimal convergence rates on numerical benchmarks. These results are substantiated by a complete mathematical analysis which provides optimal a priori error estimates.