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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
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