Discretization Methods and Iterative Solvers Based on Domain Decomposition

ISBN-13: 9783540410836 / Angielski / Miękka / 2001 / 199 str.

Barbara I. Wohlmuth; B. I. Wohlmuth

Opis

This study deals with discretization techniques on non-matching triangulations and iterative solvers with particular emphasis on mortar finite elements, Schwarz methods and multigrid techniques.

Spis treści

Discretization Techniques Based on Domain Decomposition.- 1.1 Introduction to Mortar Finite Element Methods.- 1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces.- 1.2.1 An Approximation Property.- 1.2.2 The Consistency Error.- 1.2.3 Discrete Inf-sup Conditions.- 1.2.4 Examples of Lagrange Multiplier Spaces.- 1.2.4.1 The First Order Case in 2D.- 1.2.4.2 The First Order Case in 3D.- 1.2.4.3 The Second Order Case in 2D.- 1.3 Discretization Techniques Based on the Product Space.- 1.3.1 A Dirichlet-Neumann Formulation.- 1.3.2 Variational Formulations.- 1.3.3 Algebraic Formulations.- 1.4 Examples for Special Mortar Finite Element Discretizations.- 1.4.1 The Coupling of Primal and Dual Finite Elements.- 1.4.2 An Equivalent Nonconforming Formulation.- 1.4.3 Crouzeix-Raviart Finite Elements.- 1.5 Numerical Results.- 1.5.1 Influence of the Lagrange Multiplier Spaces.- 1.5.2 A Non-optimal Mortar Method.- 1.5.3 Influence of the Choice of the Mortar Side.- 1.5.4 Influence of the Jump of the Coefficients.- Iterative Solvers Based on Domain Decomposition.- 2.1 Abstract Schwarz Theory.- 2.1.1 Additive Schwarz Methods.- 2.1.2Multiplicative Schwarz Methods.- 2.1.3 Multigrid Methods.- 2.2 Vector Field Discretizations.- 2.2.1 Raviart-Thomas Finite Elements.- 2.2.2 An Iterative Substructuring Method.- 2.2.2.1 An Interpolation Operator onto VH.- 2.2.2.2 An Extension Operator onto VF.- 2.2.2.3 Quasi-optimal Bounds.- 2.2.3 A Hierarchical Basis Method.- 2.2.3.1 Horizontal Decomposition.- 2.2.3.2 Vertical Decomposition.- 2.2.4 Numerical Results.- 2.2.4.1 The 2D Case.- 2.2.4.2 The 3D Case.- 2.3 A Multigrid Method for the Mortar Product Space Formulation.- 2.3.1 Bilinear Forms.- 2.3.2 An Approximation Property.- 2.3.3 Smoothing and Stability Properties.- 2.3.4 Implementation of the Smoothing Step.- 2.3.5 Numerical Results in 2D and 3D.- 2.3.6 Extensions to Linear Elasticity.- 2.3.6.1 Uniform Ellipticity.- 2.3.6.2 Numerical Results.- 2.3.6.3 A Weaker Interface Condition.- 2.4 A Dirichlet-Neumann Type Method.- 2.4.1 The Algorithm.- 2.4.2 Numerical Results.- 2.5 A Multigrid Method for the Mortar Saddle Point Formulation.- 2.5.1 An Approximation Property.- 2.5.2 Smoothing and Stability Properties.- 2.5.2.1 A Block Diagonal Smoother.- 2.5.2.2 An Indefinite Smoother.- 2.5.3 Numerical Results.- List of Figures.- List of Tables.- Notations.