Domain decomposition methods provide powerful and flexible tools for the numerical approximation of partial differential equations arising in the modeling of many interesting applications in science and engineering. This book deals with discretization techniques on non-matching triangulations and iterative solvers with particular emphasis on mortar finite elements, Schwarz methods and multigrid techniques. New results on non-standard situations as mortar methods based on dual basis functions and vector field discretizations are analyzed and illustrated by numerical results. The role of trace theorems, harmonic extensions, dual norms and weak interface conditions is emphasized. Although the original idea was used successfully more than a hundred years ago, these methods are relatively new for the numerical approximation. The possibilites of high performance computations and the interest in large- scale problems have led to an increased research activity.
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.
Discretization Techniques Based on Domain Decomposition.- 1.1 Introduction to Mortar Finite Element Methods.- 1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces.- 1.3 Discretization Techniques Based on the Product Space.- 1.4 Examples for Special Mortar Finite Element Discretizations.- 1.5 Numerical Results.- Iterative Solvers Based on Domain Decomposition.- 2.1 Abstract Schwarz Theory.- 2.2 Vector Field Discretizations.- 2.3 A Multigrid Method for the Mortar Product Space Formulation.- 2.4 A Dirichlet-Neumann Type Method.- 2.5 A Multigrid Method for the Mortar Saddle Point Formulation.- List of Figures.- List of Tables.- Notations.
Inhaltsverzeichnis
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.
Klappentext
Domain decomposition methods provide powerful and flexible tools for the numerical approximation of partial differential equations arising in the modeling of many interesting applications in science and engineering. This book deals with discretization techniques on non-matching triangulations and iterative solvers with particular emphasis on mortar finite elements, Schwarz methods and multigrid techniques. New results on non-standard situations as mortar methods based on dual basis functions and vector field discretizations are analyzed and illustrated by numerical results. The role of trace theorems, harmonic extensions, dual norms and weak interface conditions is emphasized. Although the original idea was used successfully more than a hundred years ago, these methods are relatively new for the numerical approximation. The possibilites of high performance computations and the interest in large- scale problems have led to an increased research activity.
Includes supplementary material: sn.pub/extras
