ExtendableFEM.jl is a high-level, extensible finite element method (FEM) library for Julia, supporting flexible problem descriptions, custom operators, and advanced grid management.

Features

• High-level, extensible API for solving PDE problems by finite element methods
• Flexible ProblemDescription interface for assigning unknowns and operators
• Supports custom kernel functions for bilinear, linear, and nonlinear forms
• Automatic assembly and Newton's method for NonlinearOperators
• Builds upon ExtendableGrids.jl and low level structures from ExtendableFEMBase.jl

Quick Example

The following minimal example demonstrates how to set up and solve a Poisson problem:

using ExtendableFEM
using ExtendableGrids # for grid management

# Build a structured ExtendableGrids.grid with mesh width h
h = 0.1
xgrid = simplexgrid(0:h:1, 0:h:1)

# Create a new PDE description
PD = ProblemDescription("Demo problem")

# Define and assign the unknown
u = Unknown("u"; name = "potential")
assign_unknown!(PD, u)

# Assign Laplace operator (diffusion term)
assign_operator!(PD, BilinearOperator([grad(u)]; factor = 1e-3))

# Assign right-hand side data
function f!(fval, qpinfo)
    x = qpinfo.x # global coordinates of quadrature point
    fval[1] = x[1] * x[2]
end
assign_operator!(PD, LinearOperator(f!, [id(u)]))

# Assign Dirichlet boundary data (u = 0) on all four grid boundary regions
assign_operator!(PD, HomogeneousBoundaryData(u; regions = [1,2,3,4]))

# Discretize: choose finite element space
FEType = H1Pk(order = 3, edim = 2, ncomponents = 1)
FES = FESpace{FEType}(xgrid)

# Solve the problem + unicode plot into terminal
sol = solve(PD, [FES]; plot = true, timeroutputs = :hide)

# Plot the solution
import PythonPlot
plot([id(u), grid(u)], sol; Plotter = PythonPlot)

Running examples from documentation

In the documentation many more examples can be found. Each of these examples is implemented as a module that needs to be included first. Afterwards the main file of the module can be run. Usually the main function has a Plotter argument that can be used to toggle a plot of the solution with an (already installed) backend of your choice (e.g. Plotter=PythonPlot, GLMakie, Plots or others supported by GridVisualize.jl). Some examples need several further dependencies. To ensure an environment were every dependency is installed, one can use the test environment via the package TestEnv. The following script runs Example201:

# activate test environment
using TestEnv
TestEnv.activate()

# include example file and load module
include("examples/Example201_PoissonProblem.jl")

# run example with default arguments
Example201_PoissonProblem.main()

# run with Plotting backend (added manually to the environment)
using PythonPlot
Example201_PoissonProblem.main(; Plotter = PythonPlot)

Citing

If you use ExtendableFEM.jl in your research, please cite this Zenodo record.

License

ExtendableFEM.jl is licensed under the MIT License. See LICENSE for details.