FEniCS.jl: Finite Element PDE Solving in Julia

FEniCS.jl is a wrapper for the FEniCS library for finite element discretizations of PDEs. This wrapper includes three parts:

Installation and direct access to FEniCS via a Conda installation. Alternatively, one may use their current FEniCS installation.
A low-level development API and provides some functionality to make directly dealing with the library a little bit easier, but still requires knowledge of FEniCS itself. Interfaces have been provided for the main functions and their attributes, and instructions to add further ones can be found here.
A high-level API for usage with DifferentialEquations. An example can be seen solving the heat equation with high order adaptive timestepping.

Various gists/jupyter notebooks have been created to provide a brief overview of the overall functionality, and of any differences between the Pythonic FEniCS and the Julian wrapper. DifferentialEquations.jl ecosystem. Paraview can also be used to visualize various results, just like in FEniCS (see below).

Installation Instructions

To get the wrapper on your system, providing a FEniCS installation exists, follow the below steps:

Add PyCall with the correct Python environment corresponding to FEniCS. Then simply add FEniCS.jl using Pkg.add("FEniCS")
Alternatively, one can install Docker and then run the following command

docker run -ti cmhyett/julia-fenics

and once inside, Julia can be accessed by calling

julia

Once inside the Julia environment, simply add FEniCS with Pkg.add("FEniCS"). All other dependencies are handled by the docker image.

This wrapper was originally started via the Google Summer of Code program along with the help of Chris Rackauckas and Bart Janssens. This was continued via GSoC '18 along with the help of Chris Rackauckas and Timo Betcke.

Tutorial

Below is a small demonstration of how a user would use our code to solve the Poisson equation with Dirichlet conditions. This directly mirrors one of the tutorials FEniCS provides

using FEniCS
mesh = UnitSquareMesh(8,8)
V = FunctionSpace(mesh,"P",1)
u_D = Expression("1+x[0]*x[0]+2*x[1]*x[1]", degree=2)
u = TrialFunction(V)
bc1 = DirichletBC(V,u_D, "on_boundary")
v = TestFunction(V)
f = Constant(-6.0)
a = dot(grad(u),grad(v))*dx
L = f*v*dx
U = FeFunction(V)
lvsolve(a,L,U,bc1) #linear variational solver
errornorm(u_D, U, norm="L2")
get_array(L) #this returns an array for the stiffness matrix
get_array(U) #this returns an array for the solution values
vtkfile = File("poisson/solution.pvd")
vtkfile << U.pyobject #exports the solution to a vtkfile

We can also plot the solution (this relies on FEniCS backend for plotting) or import it from our file into Paraview:

import PyPlot # plotting won't work if PyPlot is not imported
FEniCS.Plot(U)
FEniCS.Plot(mesh)

alt text

See the examples directory for more examples.

Reproducibility

The documentation of this SciML package was built using these direct dependencies,

Status `~/work/FEniCS.jl/FEniCS.jl/docs/Project.toml`
  [e30172f5] Documenter v1.14.1
  [186dfeec] FEniCS v0.4.2 `~/work/FEniCS.jl/FEniCS.jl`

and using this machine and Julia version.

Julia Version 1.11.6
Commit 9615af0f269 (2025-07-09 12:58 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)

A more complete overview of all dependencies and their versions is also provided.

Status `~/work/FEniCS.jl/FEniCS.jl/docs/Manifest.toml`
  [a4c015fc] ANSIColoredPrinters v0.0.1
  [1520ce14] AbstractTrees v0.4.5
  [944b1d66] CodecZlib v0.7.8
  [8f4d0f93] Conda v1.10.2
  [ffbed154] DocStringExtensions v0.9.5
  [e30172f5] Documenter v1.14.1
  [186dfeec] FEniCS v0.4.2 `~/work/FEniCS.jl/FEniCS.jl`
  [d7ba0133] Git v1.4.0
  [b5f81e59] IOCapture v0.2.5
  [92d709cd] IrrationalConstants v0.2.4
  [692b3bcd] JLLWrappers v1.7.1
  [682c06a0] JSON v0.21.4
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⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.4.3
  [92933f4c] ProgressMeter v1.10.4
  [438e738f] PyCall v1.96.4
  [2792f1a3] RegistryInstances v0.1.0
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  [44cfe95a] Pkg v1.11.0
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  [fa267f1f] TOML v1.0.3
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  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
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Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.