Introduction

Abstract

Diverse physical phenomena in such fields as biology, chemistry, metallurgy, medicine, and combustion are modelled by systems of nonlinear parabolic partial differential equations (PDEs). Nowadays there is an increasing activity in mathematics to analyse the properties of such models, including existence, uniqueness, and regularity of their solutions (e.g. Amann [6], Lunardi [107]). Due to the great complexity of such systems only little is known about true solutions. Furthermore, the permanent advance in computational capabilities allows the incorporation of more and more detailed physics into the models. Apart from a few situations, where mathematical analysis can actually be applied, the numerical analysis of PDEs is the central tool to assess the modelling process for large scale physical problems. In fact, a posteriori error estimates can be used to judge the quality of a numerical approximation and to determine an adaptive strategy to improve the accuracy where needed. In such a way numerical and modelling errors can be clearly distinguished with the effect that the reliability of the modelling process can be assessed. Moreover, successful adaptive methods lead to substantial savings in computational work for a given tolerance. They are now entering into real—life applications and starting to become a standard feature of modern software.