formoniq

A Finite Element Exterior Calculus (FEEC) engine in Rust. Partial differential equations are formulated in the language of differential forms and solved on simplicial pseudo-Riemannian manifolds of arbitrary dimension, intrinsically, without reference to any coordinate embedding.

Gradient, curl and divergence do not appear separately: they are one exterior derivative. The scalar and vector Laplacians are cases of one Hodge-Laplace operator. Dimension and form grade are runtime values, and nothing is specialized to 2D or 3D.

What it does

Coordinate-free assembly. The Galerkin operators of the Whitney discretization of the L² de Rham complex, assembled in parallel from per-cell metrics alone. An embedding, Regge signed squared edge lengths and raw metric tensors are three interchangeable geometry inputs; the solvers run identically on all three, including on manifolds with no global coordinates.
The Hodge-Laplace problem. Source and eigenvalue problems in the mixed formulation of Arnold, Falk and Winther, with the harmonic space (the cohomology of the domain) and the gauge constraint handled explicitly.
Boundary conditions as complexes. Essential conditions restrict to the relative Whitney complex, natural conditions enter as boundary loads, Robin through the boundary mass. The solvers consume any Hilbert complex unchanged.
Structure-preserving evolution. Maxwell as the Hodge-Dirac evolution on the full de Rham complex, integrated symplectically (Gauss-Legendre, energy conserved) or by an explicit Yee-style leapfrog; the Hodge heat and wave equations through the same mixed blocks.
Pure-Rust numerics. Sparse LU and Cholesky, and a shift-invert block Lanczos eigensolver for generalized pencils with singular mass blocks, via faer. No external solver toolchain.

Correctness

The test suite states the mathematics as laws and sweeps dimensions and grades: the Dirac operator squares to the negative Hodge Laplacian, energy is conserved for the hyperbolic problems and dissipates monotonically for the parabolic ones, the L² projection reproduces Whitney forms, and assembly from edge lengths agrees with assembly from metric tensors to machine precision. The examples (source, evp, heat, wave, dirac) are the end-to-end convergence and spectrum checks.

Place in the ecosystem

formoniq is the top of a small stack of standalone crates: exterior (exterior algebra), simplicial (simplicial topology and Regge geometry), glatt (the smooth continuum) and derham (discrete differential forms). See the repository for the full picture.

Origin

The first version was the BSc thesis of Luis Wirth at ETH Zürich, supervised by Prof. Dr. Ralf Hiptmair (arXiv:2506.02429).

License