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Structure and scaling in computational field theories

Final Report Summary - STUCCOFIELDS (Structure and scaling in computational field theories)

The goal of this project was to advance the branch of mathematics known as numerical analysis, in particular to introduce and analyse algorithms suitable to compute approximate solutions to partial differential equations. Structure preservation was a unifying theme, with a focus on discretizing complexes (mainly of differential forms) by subcomplexes and deriving numerical methods with a Lie group invariance. Partial differential equations can be used to model a wide variety of phenomena, and the project introduced new algorithms for fluid flow simulations, both in the singularly perturbed regime of small viscosity (convection diffusion equation) and for incompressible fluid flow (Stokes equation) and for quantum mechanical problems (computing the spectrum of Schrödinger and Dirac operators). A rigorous justification of Regge calculus, a numerical method for general relativity, was also obtained.

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.