TIEJ6801 COM2: Verification of Computational Results (JSS30) (4 op)

Nowadays, mathematical and numerical modeling has become an essential component of the general scientific process. Ever since the 1960s, numerical analysis and scientific computation have made up the most rapidly growing part of mathematics. One of the challenging problems in this area is the creation of fully reliable computer simulation methods, which could become an adequate complement to experimental sciences. This lecture course aims to give an overview of mathematical methods and computer technologies focused on reliable verification of computed solutions and present recently developed methods.

In the first week, basics on models of PDEs (partial differential equations) and numerical methods for solving them using finite difference and finite element methods will be presented. Some fast solvers for PDEs will be discussed as well. The participants will implement some of the numerical methods during the practice hours in the computer labs. The main ideas are presented in order to develop a foundation for other applications as students might work in their own research areas, as well as to be able to apply some of the techniques also in a priori and a posteriori error estimation and how it can be used for verifying computational results in practice. Methods will be discussed to determine if and how much a numerical solution of a problem is close to the exact one.

In the second week, the plan of the lecture is as follows:

• Introduction: History, Math. background, A priori error estimates.
• Error Indicators: General theory. Hierarchical, residual, averaging, and goal-oriented error indicators.
• Guaranteed error bounds for iteration methods: Difficulties and further development: variational and non-variational approaches. Functional error majorant for an ODE.
• Functional error majorants for basic elliptic PDEs: Meaning, indicators, estimates in combined error norms. Advanced forms, data oscillations.
• Guaranteed Error Bounds III: Linear elasticity. Biharmonic equation. Stokes problem. General Elliptic Problem.
• Nonlinear Models: Variational inequalities. Introduction to convex analysis. General nonlinear estimates.
• Applications: Nonconforming approximations. Modeling errors and optimal control. Problems with uncertain data. Open problems.