Final Report Summary - STUCCOFIELDS (Structure and scaling in computational field theories)
The progress was achieved by developing and analysing new finite element spaces of differential forms, in a general framework called Finite Element Systems (FES) inspired by sheaf theory. Some classical high order methods were incorporated in the FES framework, and canonical resolutions of the spaces were introduced, as a way to address the fact they these spaces do not have canonical bases. Within the FES framework, methods to obtain minimal spaces were derived, and a number of recent finite element spaces (such as serendipity spaces) were shown to be minimal spaces under given constraints. A construction of singularly perturbed FES was detailed, together with a proof of uniform stability when applied to the convection diffusion equation in the vanishing viscosity regime. For the Schrödinger and Pauli equations, a finite element method combining techniques from FES and Lattice Gauge Theory was derived and analyzed, achieving second order convergence while remaing invariant under gauge transformations. The FES framework was extended so as to encompass differential forms of higher regularity, providing new so-called Stokes complexes, relevant to incompressible materials. Finite element spaces incorporating distributions supported on subsimplices have also been developed, with applications to a posteriori error estimates. Spectral correctness of a discretization of the Dirac equation by differential forms was also proved.