A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems.

*(English)*Zbl 1059.65098A posteriori error estimates for locally mass conservative methods for subsurface flow expressed by an elliptic partial differential equations are presented. These methods are referred to as discontinuous Galerkin methods since they are based on discontinuous approximation spaces. When penalty terms are added to the bilinear form, one obtains the nonsymmetric interior penalty Galerkin methods. Formerly, the authors proved optimal rates of convergence of the methods applied to elliptic problems.

In the present paper, \(h\) adaptivity is investigated for 2D flow problems. Global explicit estimators of the \(L_2\) error are derived, and an implicit indicator of the error in the energy norm are numerically investigated. Furthermore, model problems with discontinuous coefficients are considered.

In the present paper, \(h\) adaptivity is investigated for 2D flow problems. Global explicit estimators of the \(L_2\) error are derived, and an implicit indicator of the error in the energy norm are numerically investigated. Furthermore, model problems with discontinuous coefficients are considered.

Reviewer: Fumio Kikuchi (Tokyo)

##### MSC:

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

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

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

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

35R05 | PDEs with low regular coefficients and/or low regular data |

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

76B07 | Free-surface potential flows for incompressible inviscid fluids |

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

##### Keywords:

discontinuous Galerkin method; explicit error estimators; mesh adaptivity; numerical examples; subsurface flow; nonsymmetric interior penalty Galerkin methods; convergence; elliptic problems; discontinuous coefficients
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\textit{B. Rivière} and \textit{M. F. Wheeler}, Comput. Math. Appl. 46, No. 1, 141--163 (2003; Zbl 1059.65098)

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##### References:

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