Band 12, Heft 2

This paper derives some interior superconvergence estimates for finite element solutions of the Stokes problem by a local L 2 projection method. The results depend only on local properties of the Stokes problem and the finite element approximations.

The subject-matter is the analysis of the discontinuous Galerkin finite element method of lines applied to a linear nonstationary convection–diffusion–reaction problem. In the contrary to the standard FEM the requirement of the conforming properties is omitted. The discretization is carried out with respect to space variables, whereas time remains continuous. In the discontinuous Galerkin discretization, the nonsymmetric stabilization of diffusion terms combined with interior and boundary penalty is applied. In the evaluation of fluxes the idea of upwinding is used. This allows to obtain an optimal error estimate, also verified by numerical experiments.

In this paper we propose and analyze a class of finite volume element method for solving a second order elliptic boundary value problem whose solution is defined in more than one length scales. The method has the ability to incorporate the small scale behaviors of the solution on the large scale one. This is achieved through the construction of the basis functions on each element that satisfy the homogeneous elliptic differential equation. Furthermore, the method enjoys numerical conservation feature which is highly desirable in many applications. Existing analyses on its finite element counterpart reveal that there exists a resonance error between the mesh size and the small length scale. This result motivates an oversampling technique to overcome this drawback. We develop an analysis of the proposed method under the assumption that the coefficients are of two scales and periodic in the small scale. The theoretical results are confirmed experimentally by several convergence tests. Moreover, we present an application of the method to flows in porous media.

Hp -adaptive finite element methods require error estimates of the solution at the current order and one order higher. Hierarchical-based estimation strategies have proved effective in computing errors at the current order for nonlinear parabolic equations. Recently a new approach, interpolation error-based (IEB) error estimation, for constructing a posteriori error estimates at both orders has been developed for linear reaction-diffusion equations. The main results are: (i) IEB error estimation can be applied to nonlinear reaction-diffusion equations in one space dimension; (ii) the hierarchical estimator is an implicit IEB method and thus, works for reaction-diffusion problems; (iii) a hierarchical extension for computing higher–order error estimates is asymptotically exact. Computational results illustrating the theory and comparing the implicit (hierarchical) strategy with the earlier explicit IEB methods are presented.