This library is being developed as part of the Bachelor's thesis of Luis Wirth at ETH Zurich under the supervision of Prof. Dr. Ralf Hiptmair.
Formoniq is a Rust Implementation of a Finite Element (FEM) Library based on the principles of Finite Element Extieror Calculus (FEEC) to solve partial differential equations (PDEs) formulated in terms of differential forms over simplicial manifolds using an intrinsic, coordinate-free approach.
The focus is on solving elliptic Hodge-Laplace problems with the piecewiese-linear (first-order) Whitney basis.
Finite Element Exterior Calculus (FEEC) provides a unified framework that extends the finite element method using the language of differential geometry and algebraic topology. By employing differential forms and (co-)chain complexes, FEEC offers a robust approach for preserving key topological and structural features in the solution of PDEs. This framework is particularly well-suited for problems such as the Hodge-Laplace equation and Maxwell’s equations.
Traditional finite element methods rely on explicit coordinate representations of the computational domain. However, a coordinate-free formulation aligns more naturally with the intrinsic nature of differential geometry. By representing the computational domain as a simplicial manifold with an associated Riemannian metric, we can define geometric quantities (such as lengths, areas, and volumes) intrinsically, without explicit coordinates. This metric is an inner product on the tangent spaces and defines operators like the Hodge star, which are essential in the formulation of the Hodge-Laplace operator.
Rust was chosen for its strong guarantees in memory safety, performance, and modern language features, making it ideal for high-performance computing tasks like finite elements. The Rust ownership model, borrow checker, and type system act as a proof system to ensure there are no memory bugs, race conditions, or similar undefined behaviors in any program, while achieving performance levels comparable to C/C++.