fenics-fem

FEniCS Finite Element Method for PDEs

TL;DR — Solve partial differential equations with the Finite Element Method (FEM) using FEniCS/dolfinx. Derive the weak form, generate meshes with gmsh, apply Dirichlet/Neumann boundary conditions, solve Poisson or elasticity problems, and export results to XDMF/VTK for ParaView.

When to Use

Use this Skill when you need to:

• Solve elliptic, parabolic, or hyperbolic PDEs on complex geometries
• Implement custom weak forms for multi-physics problems
• Apply mixed Dirichlet/Neumann/Robin boundary conditions
• Perform convergence studies on successively refined meshes
• Export solutions for publication-quality visualization in ParaView

Do not use this Skill when:

• You need a quick 1D finite-difference solution → use SciPy solve_bvp
• You want a spectral method for periodic domains → use pseudo-spectral libraries
• You need GPU-accelerated large-scale CFD → consider OpenFOAM or Fluidity

Background & Key Concepts

Variational (Weak) Form

The FEM converts a strong-form PDE into an integral equation by multiplying by a test function v and integrating by parts. For Poisson's equation:

Strong form: −∇²u = f in Ω, u = uD on ΓD, ∇u·n = g on ΓN

Weak form: Find u ∈ H¹(Ω) such that for all v ∈ H¹₀(Ω): ∫_Ω ∇u·∇v dx = ∫_Ω f v dx + ∫_ΓN g v ds

Function Spaces

Convergence and Error

For Lagrange P1 elements on a quasi-uniform mesh of size h:

• L² error: O(h²) (one order above approximation degree)
• H¹ error: O(h)

Environment Setup

# Recommended: use conda with conda-forge (dolfinx + gmsh are complex to compile)
conda create -n fenics-env python=3.11 -y
conda activate fenics-env
conda install -c conda-forge fenics-dolfinx mpich petsc4py gmsh pyvista -y

# Verify installation
python -c "import dolfinx; print('dolfinx version:', dolfinx.__version__)"
python -c "import gmsh; print('gmsh version:', gmsh.__version__)"

# For Docker users (simplest approach)
docker pull dolfinx/dolfinx:stable
docker run -it --rm -v $(pwd):/work dolfinx/dolfinx:stable bash