Generalization of the Zlámal condition for simplicial finite elements in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

The famous Zlámal’s minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in 2d. In this paper we present and discuss its generalization to simplicial partitions in any space dimension.     

On the equivalence of ball conditions for simplicial finite elements in Rd

We prove that the inscribed and circumscribed ball conditions, commonly used in finite element analysis, are equivalent in any dimension.     

The Cauchy–Riemann Equations: Discretization by Finite Elements,Fast Solution of the Second Variable,and a Posteriori Error Estimation

In this paper we will concentrate on the numerical solution of the Cauchy–Riemann equations. First we show that these equations bring together the finite element discretizations for the Laplace equation by standard finite elements on the one hand, and by mixed finite element methods on the other. As a consequence, methods for a posteriori error estimation for both finite element methods can derive their validity from each other. Moreover, we show that given a finite element approximation of one of the vectorfields, the missing can be accurately computed in optimal complexity.     

Numerical solution of the two-dimensional Poincaré equation

This paper deals with numerical approximation of the two-dimensional Poincaré equation that arises as a model for internal wave motion in enclosed containers. Inspired by the hyperbolicity of the equation we propose a discretisation particularly suited for this problem, which results in matrices whose size varies linearly with the number of grid points along the coordinate axes. Exact solutions are obtained, defined on a perturbed boundary. Furthermore, the problem is seen to be ill-posed and there is need for a regularisation scheme, which we base on a minimal-energy approach.     

A note on least-squares mixed finite elements in relation to standard and mixed finite elements

^** Email: brandts{at}science.uva.nl The least-squares mixed finite-element method for second-orderelliptic problems yields an approximation u_h V_h H₀¹() of thepotential u together with an approximation p_{h h} H(div ; )of the vector field p = – Au. Comparing u_h with the standardfinite-element approximation of u in V_h, and p_h with the mixedfinite-element approximation of p, it turns out that they arehigher-order perturbations of each other. In other words, theyare ‘superclose’. Refined a priori bounds and superconvergenceresults can now be proved. Also, the local mass conservationerror is of higher order than could be concluded from the standarda priori analysis.     

Demonstration of methylene-ether bridge formation in melamine-formaldehyde resins

¹H and ¹³C NMR have been used for the quantitative determination of methylene-ether bridges in melamine-formaldehyde (MF) resins. The amount of methylene-ether bridges was determined by ¹³C NMR from the number of monomethylolated amino groups consumed in the condensation reactions and is in agreement with that calculated from the condensation water contents. This latter method, which involves a combination of ¹H and ¹³C NMR, is based on the amount of condensation water released during the formation of both methylene and methylene-ether bridges. © 1995 John Wiley & Sons, Inc.     

A posteriori error estimation and adaptivity in the method of lines with mixed finite elements

We will investigate the possibility to use superconvergence results for the mixed finite element discretizations of some time-dependent partial differential equations in the construction of a posteriori error estimators. Since essentially the same approach can be followed in two space dimensions, we will, for simplicity, consider a model problem in one space dimension.