We will study Galerkin methods and its modifications for numerical discretization of parabolic PDEs. We will investigate the error estimates first for the semi-discrete problem that results from discretization in the spatial variables and then the fully discrete schemes, such as Euler and Crank-Nicolson methods. In particular, the application of semigroup theory to stability and error estimates will be demonstrated. For this reason, the survey of analytic semigroup theory will be presented. This theory will be applied in deriving maximum-norm estimates. Furthermore, the method of lumped masses will be introduced, which can be seen as a modification of the Galerkin method.  We will also consider the application of the Galerkin method in the time variable, which results in the so-called discontiunous Galerkin method.

• Thomée, Vidar. Galerkin finite element methods for parabolic problems, Vol. 25. Springer Science & Business Media, 2007
• S. Larsson. Semilinear parabolic partial differential equations - theory,approximation, and application, New Trends in the Mathematical and Computer Sciences (G. O. S. Ekhaguere, C. K. Ayo, and M. B.Olorunsaiye, eds.), Proceedings of an international workshop at Covenant University,Ota, Nigeria, June 18-23, 2006, Publications of the ICMCS, vol. 3, 2006, pp. 153-194
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Applied Math. Sciences, Vol. 44, 1983

Target audience:
This lecture is a continuation of the preceding course on "Numerical methods for partial differential equations (Numerik III)". It is intended to broaden the way towards a master thesis in the field of computational PDEs.

Prerequisites:
Participants should have some knowledge about PDEs and their numerical approximation by finite elements as provided, e.g., by the preceing course on "Numerical methods for partial differential equations (Numerik III)".

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