Finite elements III : first-order and time-dependent PDEs / Alexandre Ern, Jean-Luc Guermond.

Format:

Book

Author/Creator:

Ern, Alexandre, 1967- author.

Guermond, Jean-Luc, author.

Series:

Texts in applied mathematics ; Volume 74.

Texts in Applied Mathematics ; Volume 74

Language:

English

Subjects (All):

Calculus.

Functional analysis.

Functions.

Harmonic analysis.

Mathematical analysis.

Physical Description:

1 online resource (417 pages).

Other Title:

Finite elements 3

Finite elements three

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

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.

Contents:

Intro

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.

Notes:

Description based on print version record.

ISBN:

3-030-57348-6

OCLC:

1247665593