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Mathematical Aspects of Discontinuous Galerkin Methods / by Daniele Antonio Di Pietro, Alexandre Ern
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Mathématiques et Applications ; 69 | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2012Description: Online-Ressource (XVII, 384p, digital)ISBN:- 9783642229800
- 518.63
- 518.63 518/.63
- 518
- QA297-299.4
- QA372 .D57 2012
Contents:
Summary: Basic concepts -- Steady advection-reaction -- Unsteady first-order PDEs -- PDEs with diffusion -- Additional topics on pure diffusion -- Incompressible flows -- Friedhrichs' Systems -- Implementation.Summary: This book introduces the basic ideas for building discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. It is to a large extent self-contained and is intended for graduate students and researchers in numerical analysis. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite-element and finite-volume viewpoints are utilized to convey the main ideas underlying the design of the approximation. The analysis is presented in a rigorous mathematical setting where discrete counterparts of the key properties of the continuous problem are identified. The framework encompasses fairly general meshes regarding element shapes and hanging nodes. Salient implementation issues are also addressed.PPN: PPN: 1651096163Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Mathematical Aspects of Discontinuous Galerkin Methods; Preface; Contents; Chapter 1 Basic Concepts; 1.1 Well-Posedness for Linear Model Problems; 1.1.1 The Banach-Necas-Babuška Theorem; 1.1.2 The Lax-Milgram Lemma; 1.1.3 Lebesgue and Sobolev Spaces; 1.1.3.1 Lebesgue Spaces; 1.1.3.2 Sobolev Spaces; 1.2 The Discrete Setting; 1.2.1 The Domain ?; 1.2.2 Meshes; 1.2.3 Mesh Faces, Averages, and Jumps; 1.2.4 Broken Polynomial Spaces; 1.2.4.1 The Polynomial Space Pdk; 1.2.4.2 The Broken Polynomial Space Pdk(Th); 1.2.4.3 Other Broken Polynomial Spaces; 1.2.5 Broken Sobolev Spaces
1.2.6 The Function Space H(divO) and Its Broken Version; 1.3 Abstract Nonconforming Error Analysis; 1.3.1 The Discrete Problem; 1.3.2 Discrete Stability; 1.3.3 Consistency; 1.3.4 Boundedness; 1.3.5 Error Estimate; 1.4 Admissible Mesh Sequences; 1.4.1 Shape and Contact Regularity; 1.4.2 Geometric Properties; 1.4.3 Inverse and Trace Inequalities; 1.4.4 Polynomial Approximation; 1.4.5 The One-Dimensional Case; Part I Scalar First-Order PDEs; Chapter 2 Steady Advection-Reaction; 2.1 The Continuous Setting; 2.1.1 Assumptions on the Data; 2.1.2 The Graph Space; 2.1.3 Traces in the Graph Space
2.1.4 Weak Formulation and Well-Posedness2.1.5 Proof of Main Results; 2.1.6 Nonhomogeneous Boundary Condition; 2.2 Centered Fluxes; 2.2.1 Heuristic Derivation; 2.2.2 Error Estimates; 2.2.3 Numerical Fluxes; 2.3 Upwinding; 2.3.1 Tightened Stability Using Penalties; 2.3.2 Error Estimates Based on Coercivity; 2.3.3 Error Estimates Based on Inf-Sup Stability; 2.3.4 Numerical Fluxes; Chapter 3 Unsteady First-Order PDEs; 3.1 Unsteady Advection-Reaction; 3.1.1 The Continuous Setting; 3.1.1.1 Notation for Space-Time Functions; 3.1.1.2 Assumptions on the Data and the Exact Solution
3.1.1.3 Energy Estimate on the Exact Solution3.1.2 Space Semidiscretization; 3.1.3 Time Discretization; 3.1.4 Main Convergence Results; 3.1.4.1 Forward Euler and Finite Volume Schemes; 3.1.4.2 Explicit RK2 Schemes; 3.1.4.3 Explicit RK3 Schemes; 3.1.4.4 Numerical Illustration for Rough Solutions; 3.1.5 Analysis of Forward Euler and FiniteVolume Schemes; 3.1.5.1 The Error Equation; 3.1.5.2 Energy Identity; 3.1.5.3 Boundedness of Ahupw; 3.1.5.4 Stability; 3.1.5.5 Proof of Theorem 3.7; 3.1.6 Analysis of Explicit RK2 Schemes; 3.1.6.1 The Error Equation; 3.1.6.2 Energy Identity
3.1.6.3 Preliminary Stability Bound3.1.6.4 The Case k2: 4/3-CFL Condition; 3.1.6.5 The Piecewise Affine Case (k=1): Usual CFL Condition; 3.1.6.6 Proof of Theorem 3.10; 3.2 Nonlinear Conservation Laws; 3.2.1 The Continuous Setting; 3.2.2 Numerical Fluxes for Space Semidiscretization; 3.2.2.1 Stability; 3.2.2.2 Examples; 3.2.3 Time Discretization; 3.2.4 Limiters; 3.2.4.1 TVDM Methods; 3.2.4.2 Enforcing the TVDM Property; 3.2.4.3 Limiters in Multiple Space Dimensions; Part II Scalar Second-Order PDEs; Chapter 4 PDEs with Diffusion; 4.1 Pure Diffusion: The Continuous Setting
4.1.1 Weak Formulation and Well-Posedness
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