FEniCS

FEniCS is an open-source computing platform for solving partial differential equations using the finite element method. It emphasizes high-level problem specification, automation of variational form assembly, and efficient code generation to produce optimized solver code.

The project brings together several components: the Unified Form Language (UFL) for expressing PDE variational forms, the FEniCS Form Compiler (FFC) for generating low-level code from UFL expressions, FIAT for finite element basis functions, and DOLFIN as the problem-solving interface that handles meshes, function spaces, assembly, and linear and nonlinear solvers. FEniCS is designed for use from Python, with core performance-critical parts implemented in C++ and parallelized through MPI. It often relies on external libraries for linear algebra and solvers, such as PETSc.

Typical workflow involves defining a variational problem in UFL, using FFC to compile it to efficient code, assembling the system matrices and vectors with DOLFIN, applying boundary conditions, and solving with a suitable solver. The platform supports a range of PDEs, including Poisson, elasticity, Navier–Stokes, and convection–diffusion, on unstructured meshes and complex geometries. It also provides utilities for mesh generation, error estimation, and visualization through common tools like ParaView or VTK.

FEniCS is widely used in academic research and teaching for prototyping and solving PDEs. The project aims to provide a unified, expressive, and extensible framework that can be expanded with new elements, forms, and solvers. It maintains a community-driven development model and ongoing integration with evolving numerical methods.