This thesis focuses on developing effective r-adaptive moving mesh methods for the numerical solution of partial differential equations (PDEs). Recent develops in moving mesh methods have shown wide applicability of the methods for resolving fine scale features such as shocks and singularities in nonlinear partial differential equations. Unlike other adaptive meshing methods, the moving mesh meshing methods resolve fine scale solution behavior while keeping a fixed number of mesh points and fixed connectivity. While these methods have been successfully used to resolve features in second order PDEs, they have not been extended to higher order systems. This thesis provides the first extension of moving mesh methods to higher order operators.
The high order system modeling a capacitor structure is represented by a fourth order, semi-linear PDE and has two kinds of solution behavior: finite time singularity and sharp interface dynamics. For the singularity regime, we utilize two moving mesh techniques, an approach that satisfies the Monge-Ampére equation and an approach using a moving mesh partial differential equation (MMPDE). Both methods are used to validate and contribute to asymptotic theory that predicts the location and multiplicities of singularities in the capacitor. For interface resolution, the Monge-Ampére equation is used to generate an adaptive mesh based on the arc-length or curvature of the solution to the PDE. The methodology is capable of handling a stiff, fourth order moving interface problem with multiple fine scale solution features.
Beyond the contribution of r-adaptive moving mesh methods based on the Monge-Ampére equation for fourth order PDEs posed on rectangular domains, this thesis gives a finite difference approximation of the Monge-Ampere equation for mesh generation and adaptation for solving PDEs on curved domains. We provide a boundary mapping between a fixed, uniform rectangular domain to an adaptive, curved physical domain that is applicable to convex and select non-convex domains. We present an approach for robustly applying boundary conditions for the PDE in computational space. The effectiveness of the method is shown a variety of linear and nonlinear PDEs, including moving boundary problems, interface dynamics, and singularities.