Sergey I. Repin, Stefan A. Sauter: “Accuracy of Mathematical Models”

Abstract

The book by Repin and Sauter is an inspiring piece of work. The key question is centered around 'How accurate is a mathematical model?' by deriving model error estimates arising in different approaches for treating partial differential equations and variational problems of elliptic type. Accuracy is nothing else than measuring distances, in metric spaces or normed vector spaces, between (unknown) exact solutions and their (numerical) approximations in terms of error estimates. There is a rich theory on complete metric spaces and Banach spaces which forms the basis for some well-posedness results and numerical analysis of partial differential equations and variational problems. While the theory for linear (elliptic) problems is fairly complete, nonlinear (time-dependent) differential equations and variational inequality systems are subject of ongoing research. Error estimates often include several sources such as model-, regularization-, homogenization-, discretization-, interpolation-, linear solver-, nonlinear solver-, quadrature-, and implementation (bugs) errors. But also the exact solution to compare with, might be error-prone from inaccurate experimental setups, uncertain data, and measurement errors. The important confession one has to make is the fact that such errors never can be avoided. Rather, the aim is to design error estimates that allow us to quantify their influence and to adjust those terms with the largest contributions. B T. Wick