# Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements Source: [Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements]() Collection index: [llms.txt]() Data updated: 2026-06-05T05:01:08.426Z Authors: Andrew T. T. McRae, Colin J. Cotter, Chris J. Budd Published: 2016-12-23T20:11:40.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Mesh Movement on Swirl (test)](): Mesh Movement; Swirl (test); Error Rate (%): 37.43. - [Mesh Movement on Helmholtz (test)](): Mesh Movement; Helmholtz (test); Error Rate: 17.95. ## Original sources and implementation [Original paper]() [PDF]()