Stability Estimates for Hybrid Coupled Domain Decomposition Methods

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 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