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Special Seminar

Structure preserving approximation of nonlinear PDEs

Location:  Hill 705
Date & time: Monday, 01 February 2016 at 3:30PM - 3:31PM

Wujun Zhang, University of Maryland: Nonlinear phenomena are ubiquitous in science and engineering and are usually governed by highly nonlinear partial differential equations (PDEs). Progress of these fields to a large extent depends on progress of nonlinear analysis and PDE theory as well as numerical methods to solve the ensuing nonlinear equations. In this talk, we present structure preserving approximations to two such PDEs: nematic liquid crystals and fully nonlinear elliptic equations.

We start with the simplest one-constant model for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field \(n\) and its degree of orientation \(s\), where the pair \((s,n)\) minimizes a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a quasilinear degenerate elliptic equation for \(n\), which allows for line and plane defects to have finite energy. Our discretization with piecewise linear finite elements avoids regularization and preserves an energy inequality essential for the PDE theory to handle the degenerate elliptic part. We prove \(\Gamma\)-convergence of discrete global minimizers to continuous ones with nontrivial defects as the mesh size goes to zero, show a convergent gradient flow approach to compute minimizers, and present insightful simulations with line and plane defects. We discuss as well our future work on the \(Q\)-tensor model for liquid crystals.

In the second part of the talk, we derive a discrete Alexandroff estimate for piecewise linear functions, a novel estimate which mimics a crucial estimate in the theory of fully nonlinear elliptic PDEs. We apply this estimate to show the first rate of convergence for finite element approximations of the Monge Amp\'{e}re equation in the maximum norm. We finally discuss our future work in optimal transport problem and Gaussian curvature equations.

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Department of Mathematics

Department of Mathematics
Rutgers University
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