On error indicators for optimizing parameters in stabilized methods
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On error indicators for optimizing parameters in stabilized methods ´ s1 Petr Knobloch1 · Petr Lukaˇ
· Pavel Solin2
Received: 17 October 2018 / Accepted: 2 January 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Numerical solution of convection-dominated problems requires special techniques to suppress spurious oscillations in approximate solutions. Often, stabilized methods are applied which involve user-chosen parameters. These parameters significantly influence the quality of the solution but their optimal choice is usually not known. One possibility is to define them in an adaptive way by minimizing an error indicator characterizing the quality of the approximate solution. A non-trivial requirement on the error indicator is that its minimization with respect to the stabilization parameters should suppress spurious oscillations without smearing layers. In this paper, a new error indicator is introduced and its suitability is tested on two newly proposed benchmark problems for which previously proposed indicators do not provide satisfactory results. Mathematics Subject Classification (2010) 65N30 · 65N12
1 Introduction In this paper, we consider the scalar convection–diffusion–reaction problem ∂u ε = g on Γ N . (1) − ε u + b · ∇u + c u = f in Ω, u = u b on Γ D , ∂n Communicated by: Pavel Solin Petr Luk´asˇ
[email protected] Petr Knobloch [email protected] Pavel Solin [email protected] 1
Faculty of Mathematics and Physics, Charles University, Sokolovsk´a 49/83, 186 75 Praha 8, Prague, Czech Republic
2
University of Nevada, 1664 N Virginia St, Reno, NV 89557, USA
P. Knobloch et al.
Here, Ω ⊂ R2 is a bounded domain with a polygonal Lipschitz-continuous boundary ∂Ω and Γ D , Γ N are disjoint and relatively open subsets of ∂Ω satisfying meas1 (Γ D ) > 0 and Γ D ∪ Γ N = ∂Ω. Furthermore, let n be the outward unit normal vector to ∂Ω, ε > 0 the constant diffusivity, b ∈ W 1,∞ (Ω)2 the flow velocity, c ∈ L ∞ (Ω) the reaction coefficient, f ∈ L 2 (Ω) an outer source of u, and u b ∈ H 1/2 (Γ D ), g ∈ L 2 (Γ N ) given functions specifying the boundary conditions. We assume that the usual condition c−
1 2
div b ≥ 0
holds in Ω. Moreover, we assume that {x ∈ ∂Ω; (b · n)(x) < 0} ⊂ Γ D . In approximate solutions of problem (1) obtained using standard discretizations, spurious oscillations often appear if convection dominates diffusion. In particular, in the finite element method, a frequently used remedy is to add additional stabilization terms to the discrete problem. During the past decades, numerous stabilized methods of this type have been proposed (see, e.g., [1]) and these methods often depend on free parameters. These parameters significantly influence the quality of the approximate solution but their optimal choice is usually not known. Mostly, only their asymptotic behavior with respect to the mesh width and theoretical bounds guaranteeing wellposedness and error estimates are available. It turns out (see, e.g., [2]) that “optimal” parameters in stabilized
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