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Monge-Ampére methods for fourth order PDEs and applications to elastic interface problems

Banff International Research Station for Mathematical Innovation and Discovery

We present a robust moving mesh finite difference method simulation of fourth order nonlinear PDEs describing elastic-electrostatic interactions in two dimensions. We use and extend the Parabolic Monge-Amp Ì ere methods developed by Budd and Williams [1] to solve a fourth order PDE with finite time singularity. A key feature in our implementation is the generation of a high order transformation between computational and physical meshes that can accommodate the high order derivatives in the PDE. The PDE derived from a plate contact problem develops finite time quenching singularities at discrete spatial location(s). The moving mesh method dynamically resolves these temporally forming singularities, while preserving the underlying length scales of the problem. We show that the PMA resolves the singularities to high accuracy and gives strong evidence of self similarity near blow up. We briefly discuss the prediction of the touchdown profile using the skeleton theory from [3]. We numerically predict the skeleton set for a variety of domains, including domains with cutouts and non-convex domains. These predictions and verification of the skeleton theory are made using variational moving mesh methods developed in [2]. Accurately resolving singularities on general domains motivates recent work in extending the parabolic Monge-Amp Ì ere equation to problems with curved domains. We utilize the optimal transport methods in [4] paired with the PMA to efficiently solve PDEs on curved domains using finite difference methods on a fixed, uniform computational domain. We present preliminary results of this method for semi-linear blow-up problems, sharp interface and prescribed moving boundary problems in curved two dimensional regions. References [1] C.J. Budd and J.F. Williams. Moving mesh generation using the parabolic Monge-Ampere equation. SIAM Journal on Scientific Computing, 31(5):3438-3465, 2009. [2] W. Huang and L. Kamenski. A geometric discretization and a simple implementation for variational mesh generation and adapation. J. Compt. Phys., 301:322-337. 2015. [3] A.E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor. SIAM Journal On Applied Mathematics, 72(3):935-958, 2012. [4] J. Benamou, B. Froese, A. Oberman. Numerical solution of the optimal transportation problem using the Monge Ampere Equation. J. Compt. Phys., 260:107-126,2014.

Mathematics; Numerical analysis; Partial differential equations

video/mp4

Author affiliation: University of Notre Dame

2019-03-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69205

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/