Finite element and discontinuous Galerkin methods for transient wave equations

• Main Authors: Cohen, Gary C., 1952- (Author), Pernet, Sébastien (Author)
• Format: Electronic eBook
• Language: English
• Published: Dordrecht : Springer, [2016]
• Series: Scientific computation.
• Subjects: Finite element method.; Galerkin methods.; elektronické knihy; electronic books
• ISBN: 9789401777612; 9789401777599
• ISSN: 1434-8322
• Physical Description: 1 online resource (xvii, 381 pages) : illustrations (some color)

This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D...

Table of Contents:

• Foreword; Preface; Contents; 1 Classical Continuous Models and Their Analysis; 1.1 The Basic Equations; 1.1.1 The Acoustics Equation; 1.1.2 Maxwell's Equations; 1.1.3 The Linear Elastodynamics System; 1.1.4 Boundary Conditions; 1.2 Functional Issues; 1.2.1 Some Functional Spaces; 1.2.2 Variational Formulations; 1.2.3 Energy Identities; 1.2.4 Well-Posedness Results of Waves Equations; 1.3 Plane Wave Solutions; 1.3.1 A General Solution of the Homogeneous Wave Equation; 1.3.2 Application to Maxwell's Equations; 1.3.3 The 2D Case; 1.3.4 Application to the Isotropic Linear Elastodynamics System.
• 2.5 Tetrahedral and Triangular Edge Elements2.5.1 Mixed Formulation; 2.5.2 A First Family; 2.5.3 A Second Family; 2.5.4 Tetrahedral Mass-Lumped Edge Elements; 2.5.5 Triangular Mass-Lumped Edge Elements; 2.6 Hexahedral and Quadrilateral Edge Elements; 2.6.1 First Family; 2.6.2 Second Family; 2.7 H(div) Finite Elements; 2.7.1 Tetrahedral and Triangular Elements; 2.7.2 Hexahedral and Quadrilateral Elements; 2.8 Other Mixed Elements; 2.8.1 Pyramidal and Prismatic Edge Elements; 2.8.2 Pyramidal and Prismatic H(div) Elements; References.
• 3 Hexahedral and Quadrilateral Spectral Elements for Acoustic Waves3.1 Second-Order Formulation of the Acoustics Equation; 3.1.1 The Continuous and Approximate Problem; 3.1.2 Discretization of the Integrals; 3.2 First-Order Formulation of the Acoustics Equation; 3.2.1 The Mixed Formulation; 3.2.2 The Mass Matrices; 3.2.3 The Stiffness Matrices; 3.3 Comparison of the Methods; 3.3.1 Matrix Formulation; 3.3.2 A Theorem of Equivalence; 3.3.3 Comparison of the Costs; 3.4 Dispersion Relation; 3.4.1 The Continuous Equation; 3.4.2 A Didactic Case: The P1 Approximation.
• 3.4.3 The Concept of Numerical Dispersion3.4.4 P2 Approximation; 3.4.5 P3 and Higher-Order Approximations; 3.4.6 Extension to Higher Dimensions; 3.5 Reflection-Transmission by a Discontinuous Interface; 3.5.1 The Continuous Problem; 3.5.2 FEM Approximation of the Heterogeneous Wave Equation; 3.5.3 Taylor Expansion of the Wavenumber; 3.5.4 Interface Between Two Elements; 3.5.5 Interface at an Interior Point; 3.5.6 Extension to Higher-Order Approximations; 3.5.7 A Two-Layer Experiment; 3.6 hp-a priori Error Estimates; 3.6.1 Some Properties of Meshes.