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Grid-based approximation of partial differential equations in Julia

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# gridap/Gridap.jl

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## What

Gridap provides a set of tools for the grid-based approximation of partial differential equations (PDEs) written in the Julia programming language. The library currently supports linear and nonlinear PDE systems for scalar and vector fields, single and multi-field problems, conforming and nonconforming finite element (FE) discretizations, on structured and unstructured meshes of simplices and n-cubes. It also provides methods for time integration. Gridap is extensible and modular. One can implement new FE spaces, new reference elements, use external mesh generators, linear solvers, post-processing tools, etc. See, e.g., the list of available Gridap plugins.

Gridap has a very expressive API allowing one to solve complex PDEs with very few lines of code. The user can write the underlying weak form with a syntax almost 1:1 to the mathematical notation, and Gridap generates an efficient FE assembly loop automatically by leveraging the Julia JIT compiler. For instance, the weak form for an interior penalty DG method for the Poisson equation can be simply specified as:

```a(u,v) =
∫( ∇(v)⋅∇(u) )*dΩ +
∫( (γ/h)*v*u - v*(n_Γ⋅∇(u)) - (n_Γ⋅∇(v))*u )*dΓ +
∫(
(γ/h)*jump(v*n_Λ)⋅jump(u*n_Λ) -
jump(v*n_Λ)⋅mean(∇(u)) -
mean(∇(v))⋅jump(u*n_Λ)
)*dΛ

l(v) =
∫( v*f )*dΩ +
∫( (γ/h)*v*u - (n_Γ⋅∇(v))*u )*dΓ```

See the complete code here. As an example for multi-field PDEs, this is how the weak form for the Stokes equation with Neumann boundary conditions can be specified:

```a((u,p),(v,q)) =
∫( ∇(v)⊙∇(u) - (∇⋅v)*p + q*(∇⋅u) )*dΩ

l((v,q)) =
∫( v⋅f + q*g )*dΩ +
∫( v⋅(n_Γ⋅∇u) - (n_Γ⋅v)*p )*dΓ```

See the complete code here.

## Documentation

• STABLEDocumentation for the most recently tagged version of Gridap.jl.
• DEVELDocumentation for the in-development version of Gridap.

## Tutorials

A hands-on user-guide to the library is available as a set of tutorials. They are available as Jupyter notebooks and html pages.

## Installation

Gridap is a registered package in the official Julia package registry. Thus, the installation of Gridap is straight forward using the Julia's package manager. Open the Julia REPL, type `]` to enter package mode, and install as follows

`pkg> add Gridap`

## Examples

These are some popular PDEs solved with the Gridap library. Examples taken from the Gridap Tutorials.

## Known issues

Since Julia 1.6 onwards we have noticed large first call latencies of Gridap.jl codes with the default compiler optimization level (i.e., `-O2`). In general, while developing code, but specially if you are noting high first call latencies, we recommend to run `julia` with the `-O1` flag. For production runs use `-O2` or `-O3`.

## Gridap community

You can ask questions and interact with the Gridap community on the Julia Slack channel #gridap (see here how to join). or our gitter.

## Contributing to Gridap

Gridap is a collaborative project open to contributions. If you want to contribute, please take into account:

• Before opening a PR with a significant contribution, contact the project administrators, e.g., by writing a message in our gitter chat or by opening an issue describing what you are willing to implement. Wait for feed-back.
• Open a PR with your contribution.

Want to help? We have a number of issues waiting for help. You can start contributing to the Gridap project by solving some of those issues.

## How to cite Gridap

In order to give credit to the `Gridap` contributors, we simply ask you to cite the references below in any publication in which you have made use of the `Gridap` project. If you are using other `Gridap` sub-packages, please cite them as indicated in their repositories.

``````@article{Badia2020,
doi = {10.21105/joss.02520},
url = {https://doi.org/10.21105/joss.02520},
year = {2020},
publisher = {The Open Journal},
volume = {5},
number = {52},
pages = {2520},
author = {Santiago Badia and Francesc Verdugo},
title = {Gridap: An extensible Finite Element toolbox in Julia},
journal = {Journal of Open Source Software}
}

@article{Verdugo2022,
doi = {10.1016/j.cpc.2022.108341},
url = {https://doi.org/10.1016/j.cpc.2022.108341},
year = {2022},
month = jul,
publisher = {Elsevier {BV}},
volume = {276},
pages = {108341},
author = {Francesc Verdugo and Santiago Badia},
title = {The software design of Gridap: A Finite Element package based on the Julia {JIT} compiler},
journal = {Computer Physics Communications}
}
``````

## Contact

Grid-based approximation of partial differential equations in Julia

v0.18.2 Latest
May 3, 2024

## Packages 0

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