Functional and Numerical Analysis (FOMICS block course)

This course provides a concise introduction to some prominent subjects in functional analysis theory and its application to the analysis and numerical solution of partial differential equations (PDEs). Specifically, starting from Banach spaces and basic topological considerations, we will focus on compactness, Hilbert spaces, compact operators, the Lax-Milgram theorem, and Galerkin discretization for linear elliptic PDEs. In addition, we will consider the classical Banach fixed-point theory, and continue with more general fixed point iterations for nonlinear operators (e.g., Newton type and Kacanov schemes). Eventually, we will discuss domain decomposition methods of both overlapping and iterative substructuring types, as well as the abstract Schwarz framework, including stable decomposition estimates and interface conditions.

REFERENCES

• M. Reed, B. Simon, Functional Analysis (Methods of Modern Mathematical Physics) Academic Press, 1980
• Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext), Springer, 2010
• A. Toselli and O. B. Widlund, Domain Decomposition Methods: Algorithms and Theory, Springer, 2005.

• E. Zeidler, Nonlinear functional analysis and its Applications. II/B: Nonlinear Monotone Operators, Springer, 1990.

• E. Zeidler, Applied functional analysis. Applications to Mathematical Physics, Springer, 1995