Lecture course: Advanced Methods of Numerical Analysis 1 - 4

This lecture course is concerned with methods developed in mathematics of computations for the quantitative analysis of solutions to problems arising in various natural sciences. In last decades, these methods have seen a very rapid growth and, at the very end of the 20th century, entered a period of reformation that lasts up to now. In general, new vision and new methods arise in two main areas: (a) advanced approximation methods in the context of models described by PDE's and new (b) reliable mathematical modeling based upon explicit error control of approximate solutions and adaptive methods.

The purpose of this lecture course is to give a review of (a) and (b) methods paying a special attention to their mathematical background, connections with the methods used in mathematical physics and theory of partial differential equations for the qualitative analysis of mathematical models. On the other hand, it is intended to explain the basic principles of numerical methods used in the majority of modern computer codes and methods that can be applied to verify the accuracy of computed approximate solutions and to control errors on a given tolerance level.

First, we demonstrate basic ideas of the numerical and error control methods on the paradigm of simple linear boundary-value problems. Later, the respective generalizations are exposed for such typical engineering problems as diffusion and convection--diffusion , linear elasticity , plasticity , phase transitions in solids , and viscous flow problems . Practical training on the subject can be organized for undergraduate and Ph.D students with help of the exercises included.

AM1. Linear Models

In this part, we consider only simple elliptic and parabolic problems. The goal is to illustrate the main ideas, historical development and the state of the art in modern numerical analysis avoiding (whenever it is possible) complicated mathematical notions and argumentations. This part will also give a mathematical background and background necessary in the subsequent parts.

AM2. Elasticity theory

Main topics

(a) Physical and mathematical foundations of the elasticity theory,

(b) Dimension reduction methods,

(d) error control methods.

AM3. Theory of fluids

Main topics

(a) Mathematical models in the theory of fluids, their correctness and properties,

(b) Approximation methods and error control for Stokes, Oseen and Navier-Stokes models,

AM4. Nonlinear solid mechanics

Main topics

(a) Mathematical models associated with variational inequalities,

(b) Approximation methods and error control for elliptic problems with friction and obstacles,

(c) Approximation methods and error control in the theory of elasto-plasticity, an introduction to fracture mechanics and theory of phase transitions in solids.