Non-Linear Geometric PDEs: Algorithms, Numerical Analysis and Computation

Abstract

This dissertation presents the numerical treatment of two geometric non-linear PDEs, the Monge-Amp`ere Equation and the large bending deformation of plates under an isometry constraint. For the first problem we design a two-scale method and prove rates of convergence in the $L^\infty$ norm, which is an important progress in the numerical analysis of the Monge-Amp`ere equation and similar equations. For the second problem, we examine the deformation of the mid-plane of the plate and use the fact that it minimizes an energy functional under the isometry constraint. We design a Discontinuous Galerkin method that allows us to construct discrete minimizers of an appropriate energy functional and prove $\Gamma-$convergence to the exact minimizers. We set the theoretical foundation for this method using the problem of a single layer plate and then explore computationally the applicability of the method in more complicated and physically interesting cases using a bilayer model.