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BIRS Workshop Lecture Videos

Adaptive moving mesh method for partial differential equations DiPietro, Kelsey


In this talk, I present a robust moving mesh finite difference method for the simulation of fourth order nonlinear PDEs describing elastic-electrostatic interactions in two dimensions. The parabolic Monge-Ampà ´ere methods from [1] are extended to solve a fourth order PDE with finite time singularity. A key feature in the implementation is the generation of a high order transformation between the computational and physical meshes that can accommodate the high order derivatives in the PDE [5]. 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. I will show how the PMA resolves the singularities to high accuracy and gives strong evidence of self similarity near blow up. Next, I briefly discuss the prediction of the touchdown profile for given geometries using the skeleton theory from [3]. The skeleton set is numerically predicted for a variety of domains. The predictions of the skeleton method are verified using the variational moving mesh methods discretized in finite element space [2]. Accurately resolving singularities on general domains motivates recent work in extending the parabolic Monge-Ampà ´ere equation to problems on curved domains. Utilizing the optimal transport boundary formulation from [4, 6] creates a mapping between a fixed computational domain, on which all derivative calculations are made, to a curved physical domain. This creates an initial mesh mapping which can be paired to the PMA to adaptively resolve finescale features in the paired PDE. I give results of this method for a variety of examples including semi-linear blow-up, sharp interfaces and prescribed boundary motion on convex and select non-convex domains. [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] K. DiPietro, R. Haynes, W. Huang, A. Lindsay, Y. Yu. Moving mesh simulation of contact sets in two dimensional models of elastic-electrostatic deflection problems. J. Compt. Phys., 375:761-782, 2018. [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. [5] K. DiPietro, A. E. Lindsay. Monge-Amp\'ere simulation of fourth order PDEs in two dimensions with application to elastic-electrostatic contact problems. J. Compt. Phys.. 349:328-350, 2017. [6] B. Froese. A numerical method for the elliptic Monge-Amp\'ere equation with transport boundary conditions. SIAM Journal on Scientific Computing, 34(3):A1432-A1459, 2012.

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