fenics-fem

name	fenics-fem
description	Use this Skill to solve PDEs with the finite element method using FEniCS/dolfinx: weak form formulation, mesh generation with gmsh, Poisson/elasticity problems, boundary conditions, and paraview export.
tags	["physics","finite-element","FEniCS","PDE","numerical-methods"]
version	1.0.0
authors	[{"name":"awesome-rosetta-skills contributors","github":"@xjtulyc"}]
license	MIT
platforms	["claude-code","codex","gemini-cli","cursor"]
dependencies	{"python":["fenics-dolfinx>=0.6","gmsh>=4.11","pyvista>=0.39","numpy>=1.23","petsc4py>=3.18"]}
last_updated	2026-03-17
status	stable

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.

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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

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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

Space	dolfinx name	Use case
Continuous Galerkin deg 1	("Lagrange", 1)	Scalar fields, temperature
Continuous Galerkin deg 2	("Lagrange", 2)	Higher accuracy, elasticity displacement
Discontinuous Galerkin	("DG", 0)	Cell-wise constants, flux
Nédélec (edge elements)	("Nedelec1st", 1)	Electromagnetics, H(curl)

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)

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Environment Setup

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


python -c 
python -c 


docker pull dolfinx/dolfinx:stable
docker run -it -- -v $():/work dolfinx/dolfinx:stable bash