ExtendableFEM.jl

ExtendableFEM.jl is a high-level, extensible finite element method (FEM) library for Julia, designed for rapid prototyping and research in multiphysics and numerical analysis of FEM. It provides a flexible ProblemDescription interface, supports custom operators, and builds upon ExtendableGrids.jl and ExtendableFEMBase.jl.

Features

• High-level, modular API for PDE problems and finite element solvers
• Bilinear, linear, and nonlinear operators with custom kernel functions
• Automatic assembly and Newton's method for nonlinear operators
• Parallel assembly is possible

Dependencies

• ExtendableGrids.jl
• ExtendableSparse.jl
• ExtendableFEMBase.jl
• GridVisualize.jl
• DocStringExtensions.jl
• ForwardDiff.jl
• DiffResults.jl

Getting Started

The general workflow is as follows:

1. Geometry and Meshing

Create or load a grid using ExtendableGrids.jl:

using ExtendableGrids
# Unit square, triangulated and refined
xgrid = uniform_refine(grid_unitsquare(Triangle2D), 4)
# Uniform rectangular grid
xgrid2 = simplexgrid(0:0.1:1, 0:0.2:2)

More complex grids can be created with SimplexGridFactory.jl or by loading Gmsh files.

2. Problem Description

Pose your problem using the ProblemDescription interface:

PD = ProblemDescription()
u = Unknown("u"; name = "potential")
assign_unknown!(PD, u)
assign_operator!(PD, BilinearOperator([grad(u)]; factor = 1e-3))
f! = (result, qpinfo) -> (result[1] = qpinfo.x[1] * qpinfo.x[2])
assign_operator!(PD, LinearOperator(f!, [id(u)]))
assign_operator!(PD, HomogeneousBoundaryData(u; regions = 1:4))

3. Discretization

Choose finite element types and create spaces:

FES = FESpace{H1Pk{1,2,3}}(xgrid)
sol = FEVector(FES; tags = PD.unknowns)
fill(sol[u], 1) # optional: set initial values

4. Solve

Solve the problem:

sol = solve(PD, FES; init = sol)

For time-dependent problems, add time-derivative operators or use ODE solvers (see documentation).

5. Postprocessing and Plotting

After solving, postprocess or plot the solution:

using PyPlot
plot(id(u), sol; Plotter = PyPlot)

Gradient-Robustness

ExtendableFEM.jl supports gradient-robust schemes via reconstruction operators or divergence-free elements. Gradient-robustness ensures that discrete velocities are independent of the exact pressure in incompressible flows. See the references below for more details.

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.

References

• [1] V. John, A. Linke, C. Merdon, M. Neilan, L. Rebholz, "On the divergence constraint in mixed finite element methods for incompressible flows", SIAM Review 59(3) (2017), 492–544. Journal, Preprint
• [2] A. Linke, C. Merdon, "Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations", Computer Methods in Applied Mechanics and Engineering 311 (2016), 304–326. Journal, Preprint
• [3] M. Akbas, T. Gallouet, A. Gassmann, A. Linke, C. Merdon, "A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem", Computer Methods in Applied Mechanics and Engineering 367 (2020). Journal, Preprint