Band 14, Heft 2

In this paper, the four-parameter nonconforming finite element proposed in [30] and [19] is analyzed with the framework of Double Set Parameter (DSP) method, then it is applied to the stationary Stokes problem. The element exhibits some features of the well-known Q 1 − P 0 element under rectangular meshes. An optimal convergence rate is established for both the velocity and smoothed pressure. Furthermore, the superconvergent approximation between the interpolation of the exact solution and the finite element solution is proved. A superconvergent estimate on the centers of elements and the global superconvergence for the gradient of the velocity and the pressure are derived with the aid of a postprocessing method. Based on the superconvergence property, an asymptotically exact a posteriori estimator of ZZ type is also studied.

In this work we study a preconditioned iterative method for some higher-order time discretizations of linear parabolic partial differential equations. We use the Padé approximations of the exponential function to discretize in time and show that standard solution algorithms for lower-order time discretization schemes, such as Crank–Nicolson and implicit Euler, can be reused as preconditioners for the arising linear system. The proposed preconditioner is order optimal with respect to the discretization parameters.

In this note, we continue our studies on optimised mesh design for the Finite Element (FE) method using global norm estimates for local error control. The strategies are based on the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes (see, e.g., [3,7,18]), where error control for the primal problem is established by solving an auxiliary (dual) problem. In this context we blamed (cf. [17,18]) global norm estimates being not that useful in applications. But having a closer look at the DWR-concept, one observes that in fact global error bounds can be employed to establish local error control. We derive rigorous error bounds, especially we control the approximation process of the (unknown) dual solution entering the proposed estimate. Additional, these estimates provide information to optimise the approximation process of the primal and dual problem.

In this paper, we investigate a family of numerical methods for the approximate solution of integral equations. Here we shed light on reasons of ill posed effects and investigate several approaches to avoid those problems.