This book is the second 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 II is divided into 32 chapters plus one appendix. The first part of the volume focuses on the approximation of elliptic and mixed PDEs, beginning with fundamental results on well-posed weak formulations and their approximation by the Galerkin method. The material covered includes key results such as the BNB theorem based on inf-sup conditions, Céa's and Strang's lemmas, and the duality argument by Aubin and Nitsche. Important implementation aspects regarding quadratures, linear algebra, and assembling are also covered. The remainder of Volume II focuses on PDEs where a coercivity property is available. It investigates conforming and nonconforming approximation techniques (Galerkin, boundary penalty, Crouzeix—Raviart, discontinuous Galerkin, hybrid high-order methods). These techniques are applied to elliptic PDEs (diffusion, elasticity, the Helmholtz problem, Maxwell's equations), eigenvalue problems for elliptic PDEs, and PDEs in mixed form (Darcy and Stokes flows). Finally, the appendix addresses fundamental results on the surjectivity, bijectivity, and coercivity of linear operators in Banach spaces.
Author(s): Alexandre Ern, Jean-Luc Guermond
Series: Texts in Applied Mathematics 73
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 492
City: Cham
Tags: Finite Elements
Contents
Part V Weak formulations and well-posedness
24 Weak formulation of model problems
24.1 A second-order PDE
24.1.1 First weak formulation
24.1.2 Second weak formulation
24.1.3 Third weak formulation
24.2 A first-order PDE
24.2.1 Formulation in L1(D)
24.2.2 Formulation in L2(D)
24.3 A complex-valued model problem
24.4 Toward an abstract model problem
25 Main results on well-posedness
25.1 Mathematical setting
25.2 Lax–Milgram lemma
25.3 Banach–Nečas–Babuška (BNB) theorem
25.3.1 Test functions in reflexive Banach space
25.3.2 Test functions in dual Banach space
25.4 Two examples
25.4.1 Darcy's equations
25.4.2 First-order PDE
Part VI Galerkin approximation
26 Basic error analysis
26.1 The Galerkin method
26.2 Discrete well-posedness
26.2.1 Discrete Lax–Milgram
26.2.2 Discrete BNB
26.2.3 Fortin's lemma
26.3 Basic error estimates
26.3.1 Strong consistency: Galerkin orthogonality
26.3.2 Céa's and Babuška's lemmas
26.3.3 Approximability by finite elements
26.3.4 Sharper error estimates
27 Error analysis with variational crimes
27.1 Setting
27.2 Main results
27.2.1 The spaces Vs and V
27.2.2 Consistency/boundedness
27.2.3 Error estimate using one norm
27.2.4 Error estimate using two norms
27.2.5 Convergence
27.3 Two simple examples
27.3.1 Boundary penalty method for an elliptic PDE
27.3.2 Stabilized approximation of a first-order PDE
27.4 Strang's lemmas
27.4.1 Strang's first lemma
27.4.2 Strang's second lemma
27.4.3 Example: first-order PDE
28 Linear algebra
28.1 Stiffness and mass matrices
28.1.1 Main definitions
28.1.2 Static condensation
28.2 Bounds on the stiffness and mass matrices
28.2.1 Condition number
28.2.2 Spectrum of the mass matrix
28.2.3 Bounds on the stiffness matrix
28.2.4 Max-norm estimates
28.3 Solution methods
28.3.1 Direct methods
28.3.2 Iterative methods
29 Sparse matrices
29.1 Origin of sparsity
29.2 Storage and assembling
29.2.1 CSR and CSC formats
29.2.2 Ellpack format
29.2.3 Assembling
29.3 Reordering
29.3.1 Adjacency graph
29.3.2 Level-set ordering
29.3.3 Independent set ordering (ISO)
29.3.4 Multicolor ordering
30 Quadratures
30.1 Definition and examples
30.2 Quadrature error
30.3 Implementation
30.3.1 Nodes and weights
30.3.2 Shape functions
30.3.3 Assembling
Part VII Elliptic PDEs: conforming approximation
31 Scalar second-order elliptic PDEs
31.1 Model problem
31.1.1 Ellipticity and assumptions on the data
31.1.2 Toward a weak formulation
31.2 Dirichlet boundary condition
31.2.1 Homogeneous Dirichlet condition
31.2.2 Non-homogeneous Dirichlet condition
31.3 Robin/Neumann conditions
31.3.1 Robin condition
31.3.2 Neumann condition
31.3.3 Mixed Dirichlet–Neumann conditions
31.4 Elliptic regularity
31.4.1 Interior regularity
31.4.2 Regularity up to the boundary
32 H1-conforming approximation (I)
32.1 Continuous and discrete problems
32.2 Error analysis and best approximation in H1
32.3 L2-error analysis: the duality argument
32.3.1 Abstract duality argument
32.3.2 L2-error estimate
32.4 Elliptic projection
33 H1-conforming approximation (II)
33.1 Non-homogeneous Dirichlet conditions
33.1.1 Discrete problem and well-posedness
33.1.2 Error analysis
33.1.3 Algebraic viewpoint
33.2 Discrete maximum principle
33.3 Discrete problem with quadratures
33.3.1 Continuous and discrete settings
33.3.2 Well-posedness with quadratures
33.3.3 Error analysis with quadratures
34 A posteriori error analysis
34.1 The residual and its dual norm
34.1.1 Model problem and residual
34.1.2 The residual dual norm and the error
34.1.3 Localization of dual norms
34.2 Global upper bound
34.3 Local lower bound
34.4 Adaptivity
35 The Helmholtz problem
35.1 Robin boundary conditions
35.1.1 Well-posedness
35.1.2 A priori estimates on the solution
35.2 Mixed boundary conditions
35.3 Dirichlet boundary conditions
35.4 H1-conforming approximation
Part VIII Elliptic PDEs: nonconforming approximation
36 Crouzeix–Raviart approximation
36.1 Model problem
36.2 Crouzeix–Raviart discretization
36.2.1 Crouzeix–Raviart finite elements
36.2.2 Crouzeix–Raviart finite element space
36.2.3 Discrete problem and well-posedness
36.2.4 Discrete Poincaré–Steklov inequality
36.2.5 Bound on the jumps
36.3 Error analysis
36.3.1 Energy error estimate
36.3.2 L2-error estimate
36.3.3 Abstract nonconforming duality argument
37 Nitsche's boundary penalty method
37.1 Main ideas and discrete problem
37.2 Stability and well-posedness
37.3 Error analysis
37.3.1 Energy error estimate
37.3.2 L2-norm estimate
37.3.3 Symmetrization
38 Discontinuous Galerkin
38.1 Model problem
38.2 Symmetric interior penalty
38.2.1 Discrete problem
38.2.2 Coercivity and well-posedness
38.2.3 Variations on boundary conditions
38.3 Error analysis
38.4 Discrete gradient and fluxes
38.4.1 Liftings
38.4.2 Local formulation with fluxes
38.4.3 Equilibrated H- .4 (div) flux recovery
39 Hybrid high-order method
39.1 Local operators
39.1.1 Discrete setting
39.1.2 Local reconstruction and stabilization
39.1.3 Finite element viewpoint
39.2 Discrete problem
39.2.1 Assembling and well-posedness
39.2.2 Static condensation and global transmission problem
39.2.3 Comparison with HDG and flux recovery
39.3 Error analysis
40 Contrasted diffusivity (I)
40.1 Model problem
40.2 Discrete setting
40.3 The bilinear form n
40.3.1 Face localization of the normal diffusive flux
40.3.2 Definition of n and key identities
41 Contrasted diffusivity (II)
41.1 Continuous and discrete settings
41.2 Crouzeix–Raviart approximation
41.3 Nitsche's boundary penalty method
41.4 Discontinuous Galerkin
41.5 The hybrid high-order method
Part IX Vector-valued elliptic PDEs
42 Linear elasticity
42.1 Continuum mechanics
42.2 Weak formulation and well-posedness
42.2.1 Weak formulation
42.2.2 Korn's inequalities and well-posedness
42.3 H1-conforming approximation
42.4 Further topics
42.4.1 Crouzeix–Raviart approximation
42.4.2 Mixed finite elements
42.4.3 Hybrid high-order (HHO) approximation
43 Maxwell's equations: H- .4 (curl)-approximation
43.1 Maxwell's equations
43.1.1 The time-harmonic regime
43.1.2 The eddy current problem
43.2 Weak formulation
43.2.1 Functional setting
43.2.2 Well-posedness
43.2.3 Regularity
43.3 Approximation using edge elements
43.3.1 Discrete setting
43.3.2 H(curl)-error estimate
43.3.3 The duality argument
44 Maxwell's equations: control on the divergence
44.1 Functional setting
44.1.1 Model problem
44.1.2 A key smoothness result on the curl operator
44.2 Coercivity revisited for edge elements
44.2.1 Discrete Poincaré–Steklov inequality
44.2.2 H-.4(curl)-error analysis
44.3 The duality argument for edge elements
45 Maxwell's equations: further topics
45.1 Model problem
45.2 Boundary penalty method in H-.4(curl)
45.2.1 Discrete problem
45.2.2 Stability and well-posedness
45.2.3 Error analysis
45.3 Boundary penalty method in H1
45.4 H1-approximation with divergence control
45.4.1 A least-squares technique
45.4.2 The approximability obstruction
Part X Eigenvalue problems
46 Symmetric elliptic eigenvalue problems
46.1 Spectral theory
46.1.1 Basic notions and examples
46.1.2 Compact operators in Banach spaces
46.1.3 Symmetric operators in Hilbert spaces
46.2 Introductory examples
46.2.1 Example 1: Vibrating string
46.2.2 Example 2: Vibrating drum
46.2.3 Example 3: Stability analysis of PDEs
46.2.4 Example 4: Schrödinger equation and hydrogen atom
47 Symmetric operators, conforming approximation
47.1 Symmetric and coercive eigenvalue problems
47.1.1 Setting
47.1.2 Rayleigh quotient
47.2 H1-conforming approximation
47.2.1 Discrete setting and algebraic viewpoint
47.2.2 Eigenvalue error analysis
47.2.3 Eigenfunction error analysis
48 Nonsymmetric problems
48.1 Abstract theory
48.1.1 Approximation of compact operators
48.1.2 Application to variational formulations
48.2 Conforming approximation
48.3 Nonconforming approximation
48.3.1 Discrete formulation
48.3.2 Error analysis
Part XI PDEs in mixed form
49 Well-posedness for PDEs in mixed form
49.1 Model problems
49.1.1 Darcy
49.1.2 Stokes
49.1.3 Maxwell
49.2 Well-posedness in Hilbert spaces
49.2.1 Schur complement
49.2.2 Formulation with bilinear forms
49.2.3 Sharper a priori estimates
49.3 Saddle point problems in Hilbert spaces
49.3.1 Finite-dimensional constrained minimization
49.3.2 Lagrangian
49.4 Babuška–Brezzi theorem
49.4.1 Setting with Banach operators
49.4.2 Setting with bilinear forms and reflexive spaces
50 Mixed finite element approximation
50.1 Conforming Galerkin approximation
50.1.1 Well-posedness
50.1.2 Error analysis
50.2 Algebraic viewpoint
50.2.1 The coupled linear system
50.2.2 Schur complement
50.2.3 Augmented Lagrangian for saddle point problems
50.3 Iterative solvers
50.3.1 Uzawa algorithm
50.3.2 Krylov subspace methods
51 Darcy's equations
51.1 Weak mixed formulation
51.1.1 Dirichlet boundary condition
51.1.2 Neumann boundary condition
51.1.3 Mixed Dirichlet–Neumann boundary conditions
51.2 Primal, dual, and dual mixed formulations
51.3 Approximation of the mixed formulation
51.3.1 Discrete problem and well-posedness
51.3.2 Error analysis
52 Potential and flux recovery
52.1 Hybridization of mixed finite elements
52.1.1 From hybridization to static condensation
52.1.2 From hybridization to post-processing
52.2 Flux recovery for H1-conforming elements
52.2.1 Local flux equilibration
52.2.2 L2-norm estimate
52.2.3 Application to a posteriori error analysis
53 Stokes equations: Basic ideas
53.1 Incompressible fluid mechanics
53.2 Weak formulation and well-posedness
53.2.1 Weak formulation
53.2.2 Well-posedness
53.2.3 Regularity pickup
53.3 Conforming approximation
53.4 Classical examples of unstable pairs
53.4.1 The (mathbbQ-.41,mathbbP0) pair: Checkerboard instability
53.4.2 The (mathbbP-.41,mathbbP1) pair: Checkerboard-like instability
53.4.3 The (mathbbP-.41,mathbbP0) pair: Locking effect
54 Stokes equations: Stable pairs (I)
54.1 Proving the inf-sup condition
54.1.1 Fortin operator
54.1.2 Weak control on the pressure gradient
54.2 Mini element: the (mathbbP- .4 1-bubble,mathbbP1) pair
54.3 Taylor–Hood element: the (mathbbP- .4 2,mathbbP1) pair
54.4 Generalizations of the Taylor–Hood element
54.4.1 The (mathbbP- .4 k,mathbbPk-1) and (mathbbQ- .4 k,mathbbQk-1) pairs
54.4.2 The (mathbbP- .4 1-iso-mathbbP- .4 2,mathbbP1) and (mathbbQ- .4 1-iso-mathbbQ- .4 2,mathbbQ1) pairs
55 Stokes equations: Stable pairs (II)
55.1 Macroelement techniques
55.2 Discontinuous pressures and bubbles
55.2.1 Discontinuous pressures
55.2.2 The (mathbbP-.42,mathbbP0b) pair
55.2.3 The (mathbbP-.42-bubble,mathbbP1b) pair
55.3 Scott–Vogelius elements and generalizations
55.3.1 Special meshes
55.3.2 Stable (mathbbP-.4k,mathbbPk-1b) pairs on special meshes
55.4 Nonconforming and hybrid methods
55.5 Stable pairs with mathbbQ-.4k-based velocities
C Bijective operators in Banach spaces
References
Index
