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**Discontinuous Galerkin methods.**
*(English)*
Zbl 1036.65079

Summary: This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms.

Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to nonlinear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution.

In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.

Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to nonlinear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution.

In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.

### MSC:

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

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of 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 |

35L45 | Initial value problems for first-order hyperbolic systems |

35L65 | Hyperbolic conservation laws |