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

• Autore: Ern Alexandre <1967->
• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Descrizione fisica: 1 online resource (417 pages)
• Disciplina: 515
• Soggetto topico: Calculus; Functional analysis; Functions; Harmonic analysis; Mathematical analysis; Mètode dels elements finits; Equacions en derivades parcials
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): GuermondJean-Luc
• Nota di contenuto: 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.
• Altri titoli varianti: Finite elements 3; Finite elements three
• Titolo autorizzato: Finite elements III
• ISBN: 3-030-57348-6
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466550903316
• Lo trovi qui: Univ. di Salerno
• Serie: Texts in applied mathematics ; ; Volume 74.