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On the numerical approximation of p-biharmonic and ∞-biharmonic functions

Pryer, T. and Katzourakis, N. (2019) On the numerical approximation of p-biharmonic and ∞-biharmonic functions. Numerical Methods for Partial Differential Equations, 35 (1). pp. 155-180. ISSN 1098-2426

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To link to this item DOI: 10.1002/num.22295

Abstract/Summary

In [KP16] (arXiv:1605.07880) the authors introduced a second-order variational problem in L∞. The associated equation, coined the ∞-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by Δ2∞u:=(Δu)3|D(Δu)|2=0. In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call ∞-Biharmonic functions. For fixed p we design a mixed finite element scheme for the pre-limiting equation, the p-Bilaplacian Δ2pu:=Δ(|Δu|p−2Δu)=0. We prove convergence of the numerical solution to the weak solution of Δ2pu=0 and show that we are able to pass to the limit p→∞. We perform various tests aimed at understanding the nature of solutions of Δ2∞u and in 1-d we prove convergence of our discretisation to an appropriate weak solution concept of this problem, that of -solutions.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:77597
Publisher:Wiley

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