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Recent results prove that Nédélec edge elements do not achieve optimal rate of approximation on general quadrilateral meshes. In particular, lowest order edge elements provide stable but non-convergent approximation of Maxwell's eigenvalues. In this paper we analyze a modification of standard edge element that restores the optimality of the convergence. This modification is based on a projection technique that can be interpreted as a reduced integration procedure.

Small deformations of a viscoelastic body are considered through the linear Maxwell and Kelvin–Voigt models in the quasi-static equilibrium. A robust mixed finite element method, enforcing the symmetry of the stress tensor weakly, is proposed for these equations on simplicial tessellations in two and three dimensions. A priori error estimates are derived and numerical experiments presented. The approach can be applied to general models for linear viscoelasticity and thus offers a unified framework.

P. Galenko et al. proposed a Cahn–Hilliard model with inertial term in order to model spinodal decomposition caused by deep supercooling in certain glasses. Here we analyze a finite element space semidiscretization of their model, based on a scheme introduced by C. M. Elliott et al. for the Cahn–Hilliard equation. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates in energy norm and related norms, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity and we give a simple finite difference version of the scheme.

In this paper, we introduce new finite elements to approximate the Hellinger Reissner formulation of elasticity. The elements are the vector-valued tangential continuous Nédélec elements for the displacements, and symmetric tensor-valued, normal–normal continuous elements for the stresses. These elements do neither suffer from volume locking as the Poisson ratio approaches ½, nor suffer from shear locking when anisotropic elements are used for thin structures. We present the analysis of the new elements, discuss their implementation, and give numerical results.

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov–Poisson system. The schemes are constructed by combining a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.