Volume 9, Issue 2

A novel hybrid mixed discretisation scheme for the dual formulation of elliptic boundary value problems is presented. It is designed to facilitate the application of reliable a posteriori error estimators based on complementary energy principles. The Lagrange multipliers for the continuity constraints to be imposed on the dual variables are introduced as traces of shape functions which are of Lagrange type. Thus a conforming primal approximation can be obtained by post-processing these multipliers, while the dual approximation need no longer be conforming. The scheme may be viewed as complementing the well known equivalence of conforming dual and nonconforming primal methods. Asymptotic rates of convergence can be established for both dual and primal variables that are of optimal order.

In this paper we introduce a family of discontinuous finite element methods for fairly general second-order elliptic equations with variable coefficients. Lower-order terms are included in these equations, so the analysis and results apply also to time-dependent equations. We first write this family of methods in a mixed formulation, and then establish their equivalent versions in a nonmixed formulation by incorporating some projection operators. Within this framework, we can recover all existing discontinuous finite element methods by changing certain terms. Stability and convergence properties are studied for these discontinuous methods; stability results and sharp error estimates are established for general boundary conditions and with reasonable assumptions. We show that when discontinuous finite element methods are defined in mixed form, they not only preserve good features of these methods, they also have some advantages over classical Galerkin discontinuous methods such as they are more stable in this form.

The Arbitrary Lagrangian Eulerian formulation is widely used when problems in moving domains have to be approximated. Here we consider a time dependent linear advection–diffusion problem in a moving two-dimensional domain and discretize it by linear finite elements in space and a modification of the implicit Euler scheme, based on the mid point rule, in time. The resulting space–time discretization satisfies the so called Geometric Conservation Laws which prevent from numerical instabilities due to inaccurate calculations of geometric quantities. We derive a priori error estimates which are optimal both in space and time using slightly more regularity than that necessary for the case of non moving domains.

This paper is concerned with boundary-value problems related to the biharmonic operator. The main goal of the paper is to derive a posteriori error estimates valid for any conforming approximations of the considered problems. For this purpose, the general approach that follows from the duality theory of the calculus of variations is used. The consistency of the derived a posteriori error estimates is proved and the corresponding computational strategies are discussed.