This book is the third volume of a three-part textbook suitable for graduate coursework, professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedicated website. The salient ideas can be addressed during lecture, with the rest of the content assigned as reading material. To engage the reader, the text combines examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy.
Volume III is divided into 28 chapters. The first eight chapters focus on the symmetric positive systems of first-order PDEs called Friedrichs' systems. This part of the book presents a comprehensive and unified treatment of various stabilization techniques from the existing literature. It discusses applications to advection and advection-diffusion equations and various PDEs written in mixed form such as Darcy and Stokes flows and Maxwell's equations. The remainder of Volume III addresses time-dependent problems: parabolic equations (such as the heat equation), evolution equations without coercivity (Stokes flows, Friedrichs' systems), and nonlinear hyperbolic equations (scalar conservation equations, hyperbolic systems). It offers a fresh perspective on the analysis of well-known time-stepping methods. The last five chapters discuss the approximation of hyperbolic equations with finite elements. Here again a new perspective is proposed. These chapters should convince the reader that finite elements offer a good alternative to finite volumes to solve nonlinear conservation equations.
Author(s): Alexandre Ern, Jean-Luc Guermond
Series: Texts in Applied Mathematics 74
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 417
City: Cham
Tags: Finite Elements
Contents
Part XII First-order PDEs
56 Friedrichs' systems
56.1 Basic ideas
56.1.1 The fields mathcalK and mathcalAk
56.1.2 Integration by parts
56.1.3 The model problem
56.2 Examples
56.2.1 Advection-reaction equation
56.2.2 Darcy's equations
56.2.3 Maxwell's equations
56.3 Weak formulation and well-posedness
56.3.1 Minimal domain, maximal domain, and graph space
56.3.2 The boundary operators N and M
56.3.3 Well-posedness
56.3.4 Examples
57 Residual-based stabilization
57.1 Model problem
57.2 Least-squares (LS) approximation
57.2.1 Weak problem
57.2.2 Finite element setting
57.2.3 Error analysis
57.3 Galerkin/least-squares (GaLS)
57.3.1 Local mesh-dependent weights
57.3.2 Discrete problem and error analysis
57.3.3 Scaling
57.3.4 Examples
57.4 Boundary penalty for Friedrichs' systems
57.4.1 Model problem
57.4.2 Boundary penalty method
57.4.3 GaLS stabilization with boundary penalty
58 Fluctuation-based stabilization (I)
58.1 Discrete setting
58.2 Stability analysis
58.3 Continuous interior penalty
58.3.1 Design of the CIP stabilization
58.3.2 Error analysis
58.4 Examples
59 Fluctuation-based stabilization (II)
59.1 Two-scale decomposition
59.2 Local projection stabilization
59.3 Subgrid viscosity
59.4 Error analysis
59.5 Examples
60 Discontinuous Galerkin
60.1 Discrete setting
60.2 Centered fluxes
60.2.1 Local and global formulation
60.2.2 Error analysis
60.2.3 Examples
60.3 Tightened stability by jump penalty
60.3.1 Local and global formulation
60.3.2 Error analysis
60.3.3 Examples
61 Advection-diffusion
61.1 Model problem
61.2 Discrete setting
61.3 Stability and error analysis
61.3.1 Stability and well-posedness
61.3.2 Consistency/boundedness
61.3.3 Error estimates
61.4 Divergence-free advection
62 Stokes equations: Residual-based stabilization
62.1 Model problem
62.2 Discrete setting for GaLS stabilization
62.3 Stability and well-posedness
62.4 Error analysis
63 Stokes equations: Other stabilizations
63.1 Continuous interior penalty
63.1.1 Discrete setting
63.1.2 Stability and well-posedness
63.1.3 Error analysis
63.2 Discontinuous Galerkin
63.2.1 Discrete setting
63.2.2 Stability and well-posedness
63.2.3 Error analysis
Part XIII Parabolic PDEs
64 Bochner integration
64.1 Bochner integral
64.1.1 Strong measurability and Bochner integrability
64.1.2 Main properties
64.2 Weak time derivative
64.2.1 Strong and weak time derivatives
64.2.2 Functional spaces with weak time derivative
65 Weak formulation and well-posedness
65.1 Weak formulation
65.1.1 Heuristic argument for the heat equation
65.1.2 Abstract parabolic problem
65.1.3 Weak formulation
65.1.4 Example: the heat equation
65.1.5 Ultraweak formulation
65.2 Well-posedness
65.2.1 Uniqueness using a coercivity-like argument
65.2.2 Existence using a constructive argument
65.3 Maximum principle for the heat equation
66 Semi-discretization in space
66.1 Model problem
66.2 Principle and algebraic realization
66.3 Error analysis
66.3.1 Error equation
66.3.2 Basic error estimates
66.3.3 Application to the heat equation
66.3.4 Extension to time-varying diffusion
67 Implicit and explicit Euler schemes
67.1 Implicit Euler scheme
67.1.1 Time mesh
67.1.2 Principle and algebraic realization
67.1.3 Stability
67.1.4 Error analysis
67.1.5 Application to the heat equation
67.2 Explicit Euler scheme
67.2.1 Principle and algebraic realization
67.2.2 Stability
67.2.3 Error analysis
68 BDF2 and Crank–Nicolson schemes
68.1 Discrete setting
68.2 BDF2 scheme
68.2.1 Principle and algebraic realization
68.2.2 Stability
68.2.3 Error analysis
68.3 Crank–Nicolson scheme
68.3.1 Principle and algebraic realization
68.3.2 Stability
68.3.3 Error analysis
69 Discontinuous Galerkin in time
69.1 Setting for the time discretization
69.2 Formulation of the method
69.2.1 Quadratures and interpolation
69.2.2 Discretization in time
69.2.3 Reformulation using a time reconstruction operator
69.2.4 Equivalence with Radau IIA IRK
69.3 Stability and error analysis
69.3.1 Stability
69.3.2 Error analysis
69.4 Algebraic realization
69.4.1 IRK implementation
69.4.2 General case
70 Continuous Petrov–Galerkin in time
70.1 Formulation of the method
70.1.1 Quadratures and interpolation
70.1.2 Discretization in time
70.1.3 Equivalence with Kuntzmann–Butcher IRK
70.1.4 Collocation schemes
70.2 Stability and error analysis
70.2.1 Stability
70.2.2 Error analysis
70.3 Algebraic realization
70.3.1 IRK implementation
70.3.2 General case
71 Analysis using inf-sup stability
71.1 Well-posedness
71.1.1 Functional setting
71.1.2 Boundedness and inf-sup stability
71.1.3 Another proof of Lions' theorem
71.1.4 Ultraweak formulation
71.2 Semi-discretization in space
71.2.1 Mesh-dependent inf-sup stability
71.2.2 Inf-sup stability in the X-norm
71.3 dG(k) scheme
71.4 cPG(k) scheme
Part XIV Time-dependent Stokes equations
72 Weak formulations and well-posedness
72.1 Model problem
72.2 Constrained weak formulation
72.3 Mixed weak formulation with smooth data
72.4 Mixed weak formulation with rough data
73 Monolithic time discretization
73.1 Model problem
73.2 Space semi-discretization
73.2.1 Discrete formulation
73.2.2 Error equations and approximation operators
73.2.3 Error analysis
73.3 Implicit Euler approximation
73.3.1 Discrete formulation
73.3.2 Algebraic realization and preconditioning
73.3.3 Error analysis
73.4 Higher-order time approximation
74 Projection methods
74.1 Model problem and Helmholtz decomposition
74.2 Pressure correction in standard form
74.2.1 Formulation of the method
74.2.2 Stability and convergence properties
74.3 Pressure correction in rotational form
74.3.1 Formulation of the method
74.3.2 Stability and convergence properties
74.4 Finite element approximation
75 Artificial compressibility
75.1 Stability under compressibility perturbation
75.2 First-order artificial compressibility
75.3 Higher-order artificial compressibility
75.4 Finite element implementation
Part XV Time-dependent first-order linear PDEs
76 Well-posedness and space semi-discretization
76.1 Maximal monotone operators
76.2 Well-posedness
76.3 Time-dependent Friedrichs' systems
76.4 Space semi-discretization
76.4.1 Discrete setting
76.4.2 Discrete problem and well-posedness
76.4.3 Error analysis
77 Implicit time discretization
77.1 Model problem and space discretization
77.1.1 Model problem
77.1.2 Setting for the space discretization
77.2 Implicit Euler scheme
77.2.1 Time discrete setting and algebraic realization
77.2.2 Stability
77.3 Error analysis
77.3.1 Approximation in space
77.3.2 Error estimate in the L-norm
77.3.3 Error estimate in the graph norm
78 Explicit time discretization
78.1 Explicit Runge–Kutta (ERK) schemes
78.1.1 Butcher tableau
78.1.2 Examples
78.1.3 Order conditions
78.2 Explicit Euler scheme
78.3 Second-order two-stage ERK schemes
78.4 Third-order three-stage ERK schemes
Part XVI Nonlinear hyperbolic PDEs
79 Scalar conservation equations
79.1 Weak and entropy solutions
79.1.1 The model problem
79.1.2 Short-time existence and loss of smoothness
79.1.3 Weak solutions
79.1.4 Existence and uniqueness
79.2 Riemann problem
79.2.1 One-dimensional Riemann problem
79.2.2 Convex or concave flux
79.2.3 General case
79.2.4 Riemann cone and averages
79.2.5 Multidimensional flux
80 Hyperbolic systems
80.1 Weak solutions and examples
80.1.1 First-order quasilinear hyperbolic systems
80.1.2 Hyperbolic systems in conservative form
80.1.3 Examples
80.2 Riemann problem
80.2.1 Expansion wave, contact discontinuity, and shock
80.2.2 Maximum speed and averages
80.2.3 Invariant sets
81 First-order approximation
81.1 Scalar conservation equations
81.1.1 The finite element space
81.1.2 The scheme
81.1.3 Maximum principle
81.1.4 Entropy inequalities
81.2 Hyperbolic systems
81.2.1 The finite element space
81.2.2 The scheme
81.2.3 Upper bounds on λmax
82 Higher-order approximation
82.1 Higher order in time
82.1.1 Key ideas
82.1.2 Examples
82.1.3 Butcher tableau versus (α-β) representation
82.2 Higher order in space for scalar equations
82.2.1 Heuristic motivation and preliminary result
82.2.2 Smoothness-based graph viscosity
82.2.3 Greedy graph viscosity
83 Higher-order approximation and limiting
83.1 Higher-order techniques
83.1.1 Diminishing the graph viscosity
83.1.2 Dispersion correction: consistent mass matrix
83.2 Limiting
83.2.1 Key principles
83.2.2 Conservative algebraic formulation
83.2.3 Boris–Book–Zalesak's limiting for scalar equations
83.2.4 Convex limiting for hyperbolic systems
References
Index
