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Computational Statics and Dynamics : An Introduction Based on the Finite Element Method / / by Andreas Öchsner
| Computational Statics and Dynamics : An Introduction Based on the Finite Element Method / / by Andreas Öchsner |
| Autore | Öchsner Andreas |
| Edizione | [3rd ed. 2023.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Springer, , 2023 |
| Descrizione fisica | 1 online resource (723 pages) |
| Disciplina |
620.00151535
620.00151825 |
| Soggetto topico |
Mathematics - Data processing
Mechanics, Applied Solids Condensed matter Numerical analysis Mechanics Computational Mathematics and Numerical Analysis Solid Mechanics Condensed Matter Physics Numerical Analysis Classical Mechanics Mètode dels elements finits Mecànica Simulació per ordinador |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-09673-8 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Introduction to the Finite Element Method -- Rods and Trusses -- Euler-Bernoulli Beams and Frames -- Timoshenko Beams -- Plane Elements -- Classical Plate Elements -- Three-Dimensional Elements -- Principles of Linear Dynamics -- Integration Methods for Transient Problems -- Appendix A: Mathematics -- Appendix B: Mechanics -- Appendix C: Units and Conversion -- Appendix D: Summary of Stiffness Matrices. |
| Record Nr. | UNINA-9910659478703321 |
Öchsner Andreas
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| Cham : , : Springer International Publishing : , : Imprint : Springer, , 2023 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Finite Element Method / / Yongtao Lyu
| Finite Element Method / / Yongtao Lyu |
| Autore | Lyu Yongtao |
| Pubbl/distr/stampa | Singapore : , : Springer Nature Singapore Pte Ltd., , [2022] |
| Descrizione fisica | 1 online resource (210 pages) |
| Disciplina | 620.00151535 |
| Soggetto topico |
Finite element method
Mètode dels elements finits |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9789811933639
9789811933622 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISA-996485659803316 |
|
Lyu Yongtao
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| Singapore : , : Springer Nature Singapore Pte Ltd., , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
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Finite Element Method / / Yongtao Lyu
| Finite Element Method / / Yongtao Lyu |
| Autore | Lyu Yongtao |
| Pubbl/distr/stampa | Singapore : , : Springer Nature Singapore Pte Ltd., , [2022] |
| Descrizione fisica | 1 online resource (210 pages) |
| Disciplina | 620.00151535 |
| Soggetto topico |
Finite element method
Mètode dels elements finits |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9789811933639
9789811933622 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNINA-9910590065603321 |
|
Lyu Yongtao
|
||
| Singapore : , : Springer Nature Singapore Pte Ltd., , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Finite elements II : Galerkin approximation, elliptic and mixed PDEs / / Alexandre Ern and Jean-Luc Guermond
| Finite elements II : Galerkin approximation, elliptic and mixed PDEs / / Alexandre Ern and Jean-Luc Guermond |
| Autore | Ern Alexandre <1967-> |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (491 pages) |
| Disciplina | 515 |
| Collana | Texts in Applied Mathematics ; |
| Soggetto topico |
Calculus
Functional analysis Functions Harmonic analysis Mathematical analysis Mètode dels elements finits Equacions en derivades parcials Anàlisi funcional Mètodes de Galerkin |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-030-56923-3 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISA-996466551503316 |
|
Ern Alexandre <1967->
|
||
| Cham, Switzerland : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||
Finite elements II : Galerkin approximation, elliptic and mixed PDEs / / Alexandre Ern and Jean-Luc Guermond
| Finite elements II : Galerkin approximation, elliptic and mixed PDEs / / Alexandre Ern and Jean-Luc Guermond |
| Autore | Ern Alexandre <1967-> |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (491 pages) |
| Disciplina | 515 |
| Collana | Texts in Applied Mathematics ; |
| Soggetto topico |
Calculus
Functional analysis Functions Harmonic analysis Mathematical analysis Mètode dels elements finits Equacions en derivades parcials Anàlisi funcional Mètodes de Galerkin |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-030-56923-3 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNINA-9910483944703321 |
|
Ern Alexandre <1967->
|
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| Cham, Switzerland : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Finite elements III : first-order and time-dependent PDEs / / Alexandre Ern, Jean-Luc Guermond
| Finite elements III : first-order and time-dependent PDEs / / Alexandre Ern, Jean-Luc Guermond |
| Autore | Ern Alexandre <1967-> |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (417 pages) |
| Disciplina | 515 |
| Collana | Texts in Applied Mathematics |
| Soggetto topico |
Calculus
Functional analysis Functions Harmonic analysis Mathematical analysis Mètode dels elements finits Equacions en derivades parcials |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-030-57348-6 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| 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 |
| Record Nr. | UNISA-996466550903316 |
|
Ern Alexandre <1967->
|
||
| Cham, Switzerland : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||
Finite elements III : first-order and time-dependent PDEs / / Alexandre Ern, Jean-Luc Guermond
| Finite elements III : first-order and time-dependent PDEs / / Alexandre Ern, Jean-Luc Guermond |
| Autore | Ern Alexandre <1967-> |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (417 pages) |
| Disciplina | 515 |
| Collana | Texts in Applied Mathematics |
| Soggetto topico |
Calculus
Functional analysis Functions Harmonic analysis Mathematical analysis Mètode dels elements finits Equacions en derivades parcials |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-030-57348-6 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| 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 |
| Record Nr. | UNINA-9910484910903321 |
|
Ern Alexandre <1967->
|
||
| Cham, Switzerland : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Numerical methods for mixed finite element problems : applications to incompressible materials and contact problems / / Jean Deteix, Thierno Diop and Michel Fortin
| Numerical methods for mixed finite element problems : applications to incompressible materials and contact problems / / Jean Deteix, Thierno Diop and Michel Fortin |
| Autore | Deteix Jean |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (119 pages) |
| Disciplina | 620.00151535 |
| Collana | Lecture Notes in Mathematics |
| Soggetto topico |
Finite element method
Mètode dels elements finits |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-12616-5 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Contents -- 1 Introduction -- 2 Mixed Problems -- 2.1 Some Reminders About Mixed Problems -- 2.1.1 The Saddle Point Formulation -- 2.1.2 Existence of a Solution -- 2.1.3 Dual Problem -- 2.1.4 A More General Case: A Regular Perturbation -- 2.1.5 The Case -- 2.2 The Discrete Problem -- 2.2.1 Error Estimates -- 2.2.2 The Matricial Form of the Discrete Problem -- 2.2.3 The Discrete Dual Problem: The Schur Complement -- 2.3 Augmented Lagrangian -- 2.3.1 Augmented or Regularised Lagrangians -- 2.3.2 Discrete Augmented Lagrangian in Matrix Form -- 2.3.3 Augmented Lagrangian and the Condition Number of the Dual Problem -- 2.3.4 Augmented Lagrangian: An Iterated Penalty -- 3 Iterative Solvers for Mixed Problems -- 3.1 Classical Iterative Methods -- 3.1.1 Some General Points -- Linear Algebra and Optimisation -- Norms -- Krylov Subspace -- Preconditioning -- 3.1.2 The Preconditioned Conjugate Gradient Method -- 3.1.3 Constrained Problems: Projected Gradient and Variants -- Equality Constraints: The Projected Gradient Method -- Inequality Constraints -- Positivity Constraints -- Convex Constraints -- 3.1.4 Hierarchical Basis and Multigrid Preconditioning -- 3.1.5 Conjugate Residuals, Minres, Gmres and the Generalised Conjugate Residual Algorithm -- The Generalised Conjugate Residual Method -- The Left Preconditioning -- The Right Preconditioning -- The Gram-Schmidt Algorithm -- GCR for Mixed Problems -- 3.2 Preconditioners for the Mixed Problem -- 3.2.1 Factorisation of the System -- Solving Using the Factorisation -- 3.2.2 Approximate Solvers for the Schur Complement and the Uzawa Algorithm -- The Uzawa Algorithm -- 3.2.3 The General Preconditioned Algorithm -- 3.2.4 Augmented Lagrangian as a Perturbed Problem -- 4 Numerical Results: Cases Where Q= Q -- 4.1 Mixed Laplacian Problem -- 4.1.1 Formulation of the Problem.
4.1.2 Discrete Problem and Classic Numerical Methods -- The Augmented Lagrangian Formulation -- 4.1.3 A Numerical Example -- 4.2 Application to Incompressible Elasticity -- 4.2.1 Nearly Incompressible Linear Elasticity -- 4.2.2 Neo-Hookean and Mooney-Rivlin Materials -- Mixed Formulation for Mooney-Rivlin Materials -- 4.2.3 Numerical Results for the Linear Elasticity Problem -- 4.2.4 The Mixed-GMP-GCR Method -- Approximate Solver in u -- 4.2.5 The Test Case -- Number of Iterations and Mesh Size -- Comparison of the Preconditioners of Sect.3.2 -- Effect of the Solver in u -- 4.2.6 Large Deformation Problems -- Neo-Hookean Material -- Mooney-Rivlin Material -- 4.3 Navier-Stokes Equations -- 4.3.1 A Direct Iteration: Regularising the Problem -- 4.3.2 A Toy Problem -- 5 Contact Problems: A Case Where Q≠Q -- 5.1 Imposing Dirichlet's Condition Through a Multiplier -- 5.1.1 and Its Dual -- 5.1.2 A Steklov-Poincaré operator -- Using This as a Solver -- 5.1.3 Discrete Problems -- The Matrix Form and the Discrete Schur Complement -- 5.1.4 A Discrete Steklov-Poincaré Operator -- 5.1.5 Computational Issues, Approximate Scalar Product -- Simplified Forms of the ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S script upper P Subscript h) /StPNE pdfmark [/StBMC pdfmarkSPhps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Operator and Preconditioning -- 5.1.6 The Formulation -- The Choice of h -- 5.1.7 A Toy Model for the Contact Problem -- The Discrete Formulation -- The Active Set Strategy -- 5.2 Sliding Contact -- 5.2.1 The Discrete Contact Problem -- Contact Status -- 5.2.2 The Algorithm for Sliding Contact -- A Newton Method -- The Active Set Strategy -- 5.2.3 A Numerical Example of Contact Problem -- 6 Solving Problems with More Than One Constraint -- 6.1 A Model Problem -- 6.2 Interlaced Method -- 6.3 Preconditioners Based on Factorisation. 6.3.1 The Sequential Method -- 6.4 An Alternating Procedure -- 7 Conclusion -- Bibliography -- Index. |
| Record Nr. | UNINA-9910595041303321 |
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Deteix Jean
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| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Numerical methods for mixed finite element problems : applications to incompressible materials and contact problems / / Jean Deteix, Thierno Diop and Michel Fortin
| Numerical methods for mixed finite element problems : applications to incompressible materials and contact problems / / Jean Deteix, Thierno Diop and Michel Fortin |
| Autore | Deteix Jean |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (119 pages) |
| Disciplina | 620.00151535 |
| Collana | Lecture Notes in Mathematics |
| Soggetto topico |
Finite element method
Mètode dels elements finits |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-12616-5 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Contents -- 1 Introduction -- 2 Mixed Problems -- 2.1 Some Reminders About Mixed Problems -- 2.1.1 The Saddle Point Formulation -- 2.1.2 Existence of a Solution -- 2.1.3 Dual Problem -- 2.1.4 A More General Case: A Regular Perturbation -- 2.1.5 The Case -- 2.2 The Discrete Problem -- 2.2.1 Error Estimates -- 2.2.2 The Matricial Form of the Discrete Problem -- 2.2.3 The Discrete Dual Problem: The Schur Complement -- 2.3 Augmented Lagrangian -- 2.3.1 Augmented or Regularised Lagrangians -- 2.3.2 Discrete Augmented Lagrangian in Matrix Form -- 2.3.3 Augmented Lagrangian and the Condition Number of the Dual Problem -- 2.3.4 Augmented Lagrangian: An Iterated Penalty -- 3 Iterative Solvers for Mixed Problems -- 3.1 Classical Iterative Methods -- 3.1.1 Some General Points -- Linear Algebra and Optimisation -- Norms -- Krylov Subspace -- Preconditioning -- 3.1.2 The Preconditioned Conjugate Gradient Method -- 3.1.3 Constrained Problems: Projected Gradient and Variants -- Equality Constraints: The Projected Gradient Method -- Inequality Constraints -- Positivity Constraints -- Convex Constraints -- 3.1.4 Hierarchical Basis and Multigrid Preconditioning -- 3.1.5 Conjugate Residuals, Minres, Gmres and the Generalised Conjugate Residual Algorithm -- The Generalised Conjugate Residual Method -- The Left Preconditioning -- The Right Preconditioning -- The Gram-Schmidt Algorithm -- GCR for Mixed Problems -- 3.2 Preconditioners for the Mixed Problem -- 3.2.1 Factorisation of the System -- Solving Using the Factorisation -- 3.2.2 Approximate Solvers for the Schur Complement and the Uzawa Algorithm -- The Uzawa Algorithm -- 3.2.3 The General Preconditioned Algorithm -- 3.2.4 Augmented Lagrangian as a Perturbed Problem -- 4 Numerical Results: Cases Where Q= Q -- 4.1 Mixed Laplacian Problem -- 4.1.1 Formulation of the Problem.
4.1.2 Discrete Problem and Classic Numerical Methods -- The Augmented Lagrangian Formulation -- 4.1.3 A Numerical Example -- 4.2 Application to Incompressible Elasticity -- 4.2.1 Nearly Incompressible Linear Elasticity -- 4.2.2 Neo-Hookean and Mooney-Rivlin Materials -- Mixed Formulation for Mooney-Rivlin Materials -- 4.2.3 Numerical Results for the Linear Elasticity Problem -- 4.2.4 The Mixed-GMP-GCR Method -- Approximate Solver in u -- 4.2.5 The Test Case -- Number of Iterations and Mesh Size -- Comparison of the Preconditioners of Sect.3.2 -- Effect of the Solver in u -- 4.2.6 Large Deformation Problems -- Neo-Hookean Material -- Mooney-Rivlin Material -- 4.3 Navier-Stokes Equations -- 4.3.1 A Direct Iteration: Regularising the Problem -- 4.3.2 A Toy Problem -- 5 Contact Problems: A Case Where Q≠Q -- 5.1 Imposing Dirichlet's Condition Through a Multiplier -- 5.1.1 and Its Dual -- 5.1.2 A Steklov-Poincaré operator -- Using This as a Solver -- 5.1.3 Discrete Problems -- The Matrix Form and the Discrete Schur Complement -- 5.1.4 A Discrete Steklov-Poincaré Operator -- 5.1.5 Computational Issues, Approximate Scalar Product -- Simplified Forms of the ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S script upper P Subscript h) /StPNE pdfmark [/StBMC pdfmarkSPhps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Operator and Preconditioning -- 5.1.6 The Formulation -- The Choice of h -- 5.1.7 A Toy Model for the Contact Problem -- The Discrete Formulation -- The Active Set Strategy -- 5.2 Sliding Contact -- 5.2.1 The Discrete Contact Problem -- Contact Status -- 5.2.2 The Algorithm for Sliding Contact -- A Newton Method -- The Active Set Strategy -- 5.2.3 A Numerical Example of Contact Problem -- 6 Solving Problems with More Than One Constraint -- 6.1 A Model Problem -- 6.2 Interlaced Method -- 6.3 Preconditioners Based on Factorisation. 6.3.1 The Sequential Method -- 6.4 An Alternating Procedure -- 7 Conclusion -- Bibliography -- Index. |
| Record Nr. | UNISA-996490271503316 |
|
Deteix Jean
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| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
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