Finite element methods are not always optimal

Chebyshev Polynomials are Not Always Optimal bY

We are concerned with the problem of finding among all polynomials of degree at most n and normalized to be 1 at c the one with minimal uniform norm on 1. Here, I is a given ellipse with both foci on the real axis and c is a given real point not contained in E. Problems of this type arise in certain iterative matrix computations, and, in this context, it is generally believed and widely referen...

Piecewise linear finite element methods are not localized

Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are localized in the sense that the global dependence of pointwise errors is of higher order than the overall order of the error. These results do not indicate that such localization occurs when ...

Adaptive Finite Element Methods for Optimal

Adaptive Finite Element Methods for Optimal Control

We develop a new approach towards error control and adaptivity for nite element discretizations in optimization problems governed by partial diierential equations. Using the Lagrangian formalism the goal is to compute stationary points of the rst-order necessary opti-mality conditions. The mesh adaptation is driven by residual-based a posteriori error estimates derived by duality arguments. Thi...

Walks confined in a quadrant are not always D-finite

We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z, and always stay in the quadrant x ≥ 0, y ≥ 0. We first give a criterion which guarantees that the length generating function of these walks is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion applies, among others, to the...