Domain Decomposition Methods for Partial Differential Equations have become topics of intense interest in research over the past years because they not only provide effective computing strategies suitable for high performance computing but can provide an elegant and complete analysis of the algorithms. In these lectures, I will focus on the mathematical foundation of domain decomposition concentrating our effort on a study of lower order finite element approximation of Poisson's equation. The goal is to provide the basics of the theory in order to demonstrate what mathematical tools are required and what can be done. The topics that I plan to cover:

• Basics in Finite Elements
• Basics in Krylov-Type Methods
• Domain Decomposition Methods
• Overlapping Schwarz Methods
• Classical Schwarz Method
• Multiplicative Schwarz Method
• One-Level Additive Schwarz Algorithms
• Theoretical Bounds
• Two-level Additive Schwarz Algorithms
• Coarse Spaces
• Multilevel Algorithms (BPX)
• Schur Complement-type Methods
• Iterative Substructuring Methods
• Neumann-Neumann Methods
• Theoretical Bounds

• Restricted Schwarz Algorithms
• Symmetrized Restricted Schwarz Algorithms
• Coarse Spaces
• Overlapping Mortar Elements