Lecture 3

Approximation of Semigroups - Part 1

Dear Participants,

we all hope you survived the long Lecture 2 well. When you saw these objects (semigroups, closed operators, etc.) for the first time in your life, then it must have been a bit hard on you. But do not worry, you will have plenty of opportunities to get used to them later on.

As we have seen, time dependent partial differential equations can be considered as abstract Cauchy problems in an appropriate function space. In the most fundamental case when the equations are linear and autonomous, the solutions can be represented as semigroups. Hence, if we apply a numerical method to solve these equations, we are doing nothing else than building an approximation to our semigroup.

In this Lecture 3 we are going to understand how to formalize spatial approximations (i.e., semi-discretisations) in a unified way that is applicable to the fundamental numerical methods. There are only a few methods which do not fit into this framework and there it is then straightforward how to modify it. We will see that there are basically two cases worth studying: generator approximations (corresponding usually to finite difference schemes), and resolvent approximations (corresponding usually to Galerkin methods).

The lectures are accompanied with an appendix, where elementary space discretisation methods are briefly reviewed to motivate the abstract considerations.

We kindly ask the team of Tübingen to provide the official solutions to the exercises. However, we encourage you to send in your own solutions for discussion, as many of you have already done.

Finally, let us express our thank to you all for the great activity on the discussion pages. Typos and mistakes are pointed out and the material is being discussed in a great way. Thank you.

Best wishes and good reading,
your virtual lecturers

Alexander
András
Bálint
Petra

ISEM15_Lecture3.pdf