First-order System Least Squares (FOSLS) for Elliptic Grid Generation (EGG)

Andrea Codd

Department of Applied Mathematics
University of Colorado at Boulder
Boulder Colorado 80309-0526


Abstract

Elliptic Grid Generation (EGG), using the Winslow generator, defines a map between a simple computational region and a potentially complicated physical region. It can be used numerically to create a mesh for use with a finite element or other discretization method to solve the system of equations posed on the physical domain. Alternatively, it can be used to transform equations posed on the physical region to a logical region, where the transformed equations are then solved. EGG allows complete specification of the boundary maps. Moreover, by choosing the computational region to be convex, we can ensure that the Jacobian of the map is positive, and that the map is one-to-one and onto.

Here we explore a new numerical method for solving the Winslow EGG equations. The approach is based on first-order system least squares (FOSLS) for formulating the problem, Newton's method for linearization, and algebraic multigrid for the matrix solver. The basic purpose is to provide a fully variational principle for EGG that facilitates accurate discretization and fast solution methods. We develop theoretical estimates and demonstrate the potential of the method with numerical experiments.