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A robust a posteriori error estimate for hp-adaptive DG methods for convection-diffusion equations. (English) Zbl 1225.65104

The authors consider a model for convection-diffusion equations in two dimensions. They derive a robust a posteriori error estimate for \(h_p\)-adaptive discontinuous Galerkin (DG) methods for the problem under consideration. The ratio of the constants in the reliability and efficiency bounds is independent of the Péclet number \(\varepsilon\) of the equation, and hence the estimate is fully robust. The estimates are applied as an error indicator in an \(h_p\)-adaptive refinement algorithm. Numerical examples show that the indicator is effective in locating and resolving the interior and boundary layers. Once the local mesh size is of the same order as the width of the boundary or interior layer, both the energy error and the error indicator are observed to convergence exponentially.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

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