Michal Křížek

Bio: Michal Křížek is an academic researcher from Academy of Sciences of the Czech Republic. The author has contributed to research in topics: Finite element method & Mixed finite element method. The author has an hindex of 21, co-authored 131 publications receiving 1966 citations. Previous affiliations of Michal Křížek include University of Amsterdam & Czechoslovak Academy of Sciences.

On superconvergence techniques

Abstract: A brief survey with a bibliography of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented. A particular emphasis is laid on superconvergent schemes for elliptic problems in the plane employing the finite element method.

On the maximum angle condition for linear tetrahedral elements

Abstract: Synge’s maximum angle condition for triangular elements is generalized to tetrahedral elements. For the generalized condition, it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains $O(h)$ in the $W_p^1 (\Omega )$-norm for sufficiently smooth functions and $p \in [1,\infty ]$.

Superconvergence phenomenon in the finite element method arising from averaging gradients

Abstract: We study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements. The averaged gradient is a piecewise linear continuous vector field, the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point. The convergence rate of the averaged gradient to an exact gradient in theL 2-norm can locally be higher even by one than that of the original piecewise constant discrete gradient.

On Nonobtuse Simplicial Partitions

Abstract: This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

A Posteriori Error Estimation in Finite Element Analysis

Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

Abstract: This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

The P and H-P versions of the finite element method, basic principles and properties

Abstract: In the classical form of the finite element method called the hversion, piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy. In this paper, we discuss the fundamental theoretical ideas behind the relatively recent p version and h-p version. In the p version, a fixed mesh is used and p is allowed to increase. The h-p version combines both approaches. The authors describe and explain the basic properties and characteristics of these newer versions, especially in areas where their behavior is significantly different from that of the h version. Simplified proofs of key concepts are included and computational illustrations of several results are provided. A benchmark comparison between the various versions in included.