Abstract
We give a relatively complete analysis of the penalty methods for one-sided parabolic problems with piecewise smooth obstacles, which include an important subclass of polyhedra. It is shown that the original variational inequality can be equivalently formulated as unconstrained variational inequality with a locally Lipschitz functional, which has a different form for convex and non-convex obstacles. The regularity of the solution of the problem in the case of convex obstacles is investigated. The penalty method is obtained by regularizing a non-differentiable functional by a differentiable one. In comparison with the known methods, the penalty operator obtained in this way in the case of non-convex obstacles includes an additional (less regular) term. The accuracy of the constructed penalty methods is investigated in uniform and energy norms. The additional term in the penalty operator made it possible to obtain accuracy estimates of the same order as in the case of smooth obstacles.
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Notes
They often talk about the regularization method if the operator \(\beta_{\varepsilon}\) is uniformly bounded over \(\varepsilon\) and about the penalty method otherwise.
In [10], the case \(\psi=0\) was considered. Under conditions (2) inequality (1) is reduced to this case by the shifts \(u\to u-\psi\) and \(f\to f+\triangle\psi\).
Nothing prevents the consideration of the case when in each integral in (16) its own function \(\Theta\) of the form (15) is chosen.
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Funding
This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program and by Russian Foundation for Basic Research, project no. 19-01-00431.
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Dautov, R.Z. Penalty Methods for One-Sided Parabolic Problems with Piecewise Smooth Obstacles. Lobachevskii J Math 42, 1643–1651 (2021). https://doi.org/10.1134/S1995080221070064
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DOI: https://doi.org/10.1134/S1995080221070064