On the numerical approximation of $p$-Biharmonic and $\infty$-Biharmonic functions

Abstract

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

23 pages, 5 figures

References in corpus (2)

• Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian
• An a posteriori analysis of some inconsistent, nonconforming Galerkin methods approximating elliptic problems

Cited by in corpus (3)

• Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian
• Phase separation of n dimensional infinity Harmonic mappings
• A Symmetry Analysis of the $\infty$-Polylaplacian