Séance Séminaire

NEMESIS Seminar

A Small-Simplices Framework for Explicit Potentials and Discrete Poincaré–Friedrichs Estimates

We develop a discrete framework for computable estimates of Poincaré–Friedrichs constants for trimmed polynomial differential forms on simplicial meshes. The construction is based on explicit potential operators, viewed as right inverses of the exterior derivative with controlled norms. To obtain them, we revisit the small-simplices approach through a quotient-complex formulation that preserves the differential structure while retaining geometrically meaningful degrees of freedom. This yields explicit reconstruction maps between cochain spaces carried by small simplices and trimmed polynomial finite element spaces. A key feature of the method is that it provides a single potential operator on the whole simplicial patch and does not rely on shellability assumptions. The resulting operator admits a natural factorization into interpolation, discrete potential, and reconstruction steps, which separates topological and metric contributions and leads to computable upper bounds for the associated discrete Poincaré–Friedrichs constants. In the Hilbert case, these bounds reduce to finite-dimensional spectral quantities. Numerical experiments in two and three space dimensions illustrate the construction and the influence of the chosen simplicial realization on the quality of the resulting estimates.