GridapGeosciences.jl: A Julia finite element package for partial differential equations on general manifolds
Abstract
We present GridapGeosciences$.$jl, a new parallel distributed-memory Julia package for the numerical approximation of partial differential equations on general manifolds. Our abstract approach to generating discrete manifold domains relies on mesh refinement in general, and is implemented within the Gridap finite element library, whose high-level interface enables seamless definition of both intrinsic and extrinsic variational formulations. GridapGeosciences$.$jl features the cubed sphere manifold, which is used in this study to showcase the functionality of GridapGeosciences$.$jl in different applications. The first application involves scalar transport where we develop intrinsic formulations of common stabilisation terms, which are shown to be equivalent to well-known extrinsic stabilisation formalisations. Such stabilisation terms are used to develop an intrinsic formulation of the two-dimensional thermal shallow water equations that includes upwinded numerical fluxes. The final application is a linearisation of the Boussinesq equations where we demonstrate the capacity of GridapGeosciences$.$jl to capture both surface and vertical dynamics on a three-dimensional manifold.
Related
- An intrinsic finite element framework for scalar- and vector-valued partial differential equations on general manifolds
- Differential Harnack Estimates and Frequency Monotonicity for Filtration Equations on Riemannian Manifolds
- Bounds-Constrained Finite Element Approximation of Time-Dependent Partial Differential Equations
- Rough differential equations on manifolds via natural bundles
- Linking numbers and special differential characters in arithmetic hyperbolic manifolds
BibTeX
@misc{tambyah2026gridapgeosciencesjl,
title = {{GridapGeosciences.jl: A Julia finite element package for partial differential equations on general manifolds}},
author = {Tamara A. Tambyah and Alberto F. Martín and David Lee and Santiago Badia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08036},
eprint = {2609.08036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08036},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - GridapGeosciences.jl: A Julia finite element package for partial differential equations on general manifolds
AU - Tamara A. Tambyah
AU - Alberto F. Martín
AU - David Lee
AU - Santiago Badia
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.08036
UR - https://arxiv.org/abs/2609.08036
AB - We present GridapGeosciences$.$jl, a new parallel distributed-memory Julia package for the numerical approximation of partial differential equations on general manifolds. Our abstract approach to generating discrete manifold domains relies on mesh refinement in general, and is implemented within the Gridap finite element library, whose high-level interface enables seamless definition of both intrinsic and extrinsic variational formulations. GridapGeosciences$.$jl features the cubed sphere manifold, which is used in this study to showcase the functionality of GridapGeosciences$.$jl in different applications. The first application involves scalar transport where we develop intrinsic formulations of common stabilisation terms, which are shown to be equivalent to well-known extrinsic stabilisation formalisations. Such stabilisation terms are used to develop an intrinsic formulation of the two-dimensional thermal shallow water equations that includes upwinded numerical fluxes. The final application is a linearisation of the Boussinesq equations where we demonstrate the capacity of GridapGeosciences$.$jl to capture both surface and vertical dynamics on a three-dimensional manifold.
ER -