Domain decomposition methods are a well established tool for an efficient numerical solution of partial differential equations, in particular for the coupling of different model equations and of different discretization methods. Based on the approximate solution of local boundary value problems either by finite or boundary element methods, the global problem is reduced to an operator equation on the skeleton of the domain decomposition. Different variational formulations then lead to hybrid domain decomposition methods.

Introduction

1. Preliminaries
   1.1 Sobolev Spaces
   1.2 Saddle Point Problems
   1.3 Finite Element Spaces
   1.4 Projection Operators
   1.5 Quasi Interpolation Operators
2. Stability Results
   2.1 Piecewise Linear Elements
   2.2 Dual Finite Element Spaces
   2.3 Higher Order Finite Element Spaces
   2.4 Biorthogonal Basis Functions
3. The Dirichlet-Neumann Map for Elliptic Boundary Value Problems
   3.1 The Steklov-Poincare Operator
   3.2 The Newton Potential
   3.3 Approximation by Finite Element Methods
   3.4 Approximation by Boundary Element Methods
4. Mixed Discretization Schemes
   4.1 Variational Methods with Approximate Steklov-Poincare Operators
   4.2 Lagrange Multiplier Methods
5. Hybrid Coupled Domain Decomposition Methods
   5.1 Dirichlet Domain Decomposition Methods
   5.2 A Two-Level Method
   5.3 Three-Field Methods
   5.4 Neumann Domain Decomposition Methods
   5.5 Numerical Results
   5.6 Concluding Remarks
   References