Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements

TL;DR

This paper develops a finite element method for generating adaptive meshes on the plane and sphere based on optimal transport and Monge-Ampère equations, with numerical validation using Firedrake.

Contribution

It extends finite element schemes for Monge-Ampère equations to mesh adaptivity on both the plane and sphere, incorporating nonlinearities from monitor functions.

Findings

Effective mesh adaptation on plane and sphere demonstrated.

Finite element scheme successfully implemented in Firedrake.

Numerical examples validate the approach.

Abstract

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal transport condition, this leads to a Monge-Amp\`ere equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge-Amp\`ere equation to this mesh generation problem; this is a mixed finite element scheme, in which an extra discrete variable is introduced to represent the Hessian matrix of second derivatives. The problem we consider has additional nonlinearities over the basic Monge-Amp\`ere equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive the equivalent Monge-Amp\`ere-like equation for generating meshes on the sphere. The finite element scheme is extended to the sphere, and we provide numerical examples. All numerical experiments are performed using the open-source finite element framework Firedrake.