### 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 an 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.

Original language | English |
---|---|

Pages (from-to) | A1121-A1148 |

Journal | SIAM Journal on Scientific Computing |

Volume | 40 |

Issue number | 2 |

Early online date | 24 Apr 2018 |

DOIs | |

Publication status | Published - 24 Apr 2018 |

### Fingerprint

### Keywords

- Finite element
- Mesh adaptivity
- Monge-Amp ere equation
- Optimal transport

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*40*(2), A1121-A1148. https://doi.org/10.1137/16M1109515

**Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements.** / McRae, Andrew T.T.; Cotter, Colin J.; Budd, Chris J.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 40, no. 2, pp. A1121-A1148. https://doi.org/10.1137/16M1109515

}

TY - JOUR

T1 - Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements

AU - McRae, Andrew T.T.

AU - Cotter, Colin J.

AU - Budd, Chris J.

PY - 2018/4/24

Y1 - 2018/4/24

N2 - 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 an 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.

AB - 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 an 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.

KW - Finite element

KW - Mesh adaptivity

KW - Monge-Amp ere equation

KW - Optimal transport

UR - http://www.scopus.com/inward/record.url?scp=85046714208&partnerID=8YFLogxK

U2 - 10.1137/16M1109515

DO - 10.1137/16M1109515

M3 - Article

VL - 40

SP - A1121-A1148

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 2

ER -