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Robust hierarchical a posteriori error estimators for stabilized convection-diffusion problems. (English) Zbl 1254.65122

The authors construct and analyze a posteriori error estimators for the convection-diffusion problem: \(-\epsilon \Delta u + b\cdot\nabla u + \sigma u = f\) in \(\Omega\subset {\mathbb R}^d\), with zero boundary condition on \(\partial \Omega\), where \(f \in L^2(\Omega)\), \(\sigma\in L^\infty(\Omega)\) and \(b\in(W^{1,\infty}(\Omega))^d\). It is assumed that there are two positive constants \(c_1\) and \(c_2\) not depending on \(\epsilon\) such that \(\|\sigma\|_\infty\leq c_2\) and \(-(1/2)\,{\text{div}}\,b+\sigma\geq c_1\). This estimator of hierarchical type recovers robustness. By an appropriate choice of the finite element spaces and norms, the estimators yield an upper and lower bounds of the error. The estimator is robust with respect to the physical parameters of the problem. Specifically, this a posteriori analysis for the stabilized convection-diffusion problem establishes that the error is bounded uniformly in \(\epsilon\). The main results show that the estimator is efficient and robust.

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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