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A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. (English) Zbl 1059.65098
A 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.

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
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