Discontinuous Galerkin methods for solving elliptic and parabolic equations. Theory and implementation.

*(English)*Zbl 1153.65112
Frontiers in Applied Mathematics 35. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-56-6/pbk; 978-0-89871-744-0/ebook). xxii, 190 p. (2008).

One main aim of the book is to present basic tools for analyzing discontinuous Galerkin (DG) methods. Proofs of stability and convergence are given in detail. Another objective is to make the reader familiar with coding issues of DG methods such as data structures, construction of local matrices, and assembling of the global matrix. Several nontrivial computational examples for engineering problems are provided.

Chapters 1-4 are concerned with the general presentation of the DG method and its analysis. The DG method is nonconforming, i.e. test spaces (of polynomials) are used that are globally discontinuous and are not elements of the solution space as in the case of the conforming Galerkin method. The discontinuities along the edges or faces of the underlying domain triangulation lead to jump contributions, so-called penalty terms, in the variational formulation of the DG method. There are different ways to take the penalty terms into account leading to the symmetric interior penalty Galerkin method (SIPG), nonsymmetric interior penalty Galerkin method (NIPG) and the incomplete interior penalty Galerkin method (IIPG), respectively.

The DG method is introduced for one-dimensional elliptic problems in Chapter 1 and then presented for higher dimensional linear elliptic problems in Chapter 2, where also the analysis is given. Error estimates are derived in the energy and the \(L^2\) norm. Details of the implementation of the DG method and numerical experiments are provided. Chapter 3 is concerned with the DG method applied to linear parabolic problems including time continuous and various time discrete formulations. In the following chapter the same program is done for parabolic equations with convection using an upwind discretization of the convection term. Also in these two chapters a convergence analysis is included.

The last 52 pages are dedicated to the application of the DG method to linear elasticity, Stokes flow, Navier-Stokes flow, flow in porous media.

The text contains brief definitions of the function spaces entering in the analysis of the DG method, and some further tools needed are cited. In these parts one could think of some improvements. In the definition of the spaces \(H^s(\Omega)\) and \(H^s(\partial\Omega)\) for polynomial domains \(\Omega\) problems arise for \(s \geq 3/2\) as well as in the application of Green’s formula. The continuity of the trace operators and not only inclusion properties should be stated. In the definition of the \(L^p(0,T;V)\) spaces Bochner integrals are required not just measurable functions only. Inverse inequalities depend on the geometry of the underlying domain and cannot be so simply stated as in Section 3.1.5.

The present book can be recommended for readers interested to get a good introduction to the theoretical and numerical fundamentals of the DG method and its possible applications.

Chapters 1-4 are concerned with the general presentation of the DG method and its analysis. The DG method is nonconforming, i.e. test spaces (of polynomials) are used that are globally discontinuous and are not elements of the solution space as in the case of the conforming Galerkin method. The discontinuities along the edges or faces of the underlying domain triangulation lead to jump contributions, so-called penalty terms, in the variational formulation of the DG method. There are different ways to take the penalty terms into account leading to the symmetric interior penalty Galerkin method (SIPG), nonsymmetric interior penalty Galerkin method (NIPG) and the incomplete interior penalty Galerkin method (IIPG), respectively.

The DG method is introduced for one-dimensional elliptic problems in Chapter 1 and then presented for higher dimensional linear elliptic problems in Chapter 2, where also the analysis is given. Error estimates are derived in the energy and the \(L^2\) norm. Details of the implementation of the DG method and numerical experiments are provided. Chapter 3 is concerned with the DG method applied to linear parabolic problems including time continuous and various time discrete formulations. In the following chapter the same program is done for parabolic equations with convection using an upwind discretization of the convection term. Also in these two chapters a convergence analysis is included.

The last 52 pages are dedicated to the application of the DG method to linear elasticity, Stokes flow, Navier-Stokes flow, flow in porous media.

The text contains brief definitions of the function spaces entering in the analysis of the DG method, and some further tools needed are cited. In these parts one could think of some improvements. In the definition of the spaces \(H^s(\Omega)\) and \(H^s(\partial\Omega)\) for polynomial domains \(\Omega\) problems arise for \(s \geq 3/2\) as well as in the application of Green’s formula. The continuity of the trace operators and not only inclusion properties should be stated. In the definition of the \(L^p(0,T;V)\) spaces Bochner integrals are required not just measurable functions only. Inverse inequalities depend on the geometry of the underlying domain and cannot be so simply stated as in Section 3.1.5.

The present book can be recommended for readers interested to get a good introduction to the theoretical and numerical fundamentals of the DG method and its possible applications.

Reviewer: Rolf Dieter Grigorieff (Berlin)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65N40 | Method of lines for boundary value problems involving PDEs |

35Q30 | Navier-Stokes equations |

76S05 | Flows in porous media; filtration; seepage |

76M10 | Finite element methods applied to problems in fluid mechanics |

74B05 | Classical linear elasticity |

74S05 | Finite element methods applied to problems in solid mechanics |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35K15 | Initial value problems for second-order parabolic equations |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76D07 | Stokes and related (Oseen, etc.) flows |