Tore Gunnar Halvorsen

A simplicial gauge theory

We provide an action for gauge theories discretized on simplicial meshes, inspired by lattice gauge theory and finite element methods. The action is discretely gauge invariant and we give a proof of consistency. A discrete Noethers theorem that can be applied to our setting, is also proved.

A Gauge Invariant Discretization on Simplicial Grids of the Schrödinger Eigenvalue Problem in an Electromagnetic Field

We propose a method to compute approximate eigenpairs of the Schrodinger operator on a bounded domain in the presence of an electromagnetic field. Formulated for simplicial meshes, the method combines techniques from lattice gauge theory and finite elements, retaining the discrete gauge invariance of the former but allowing for noncongruent space elements as in the latter. The error of the method is studied in the framework of Strangs variational crimes, comparing with a standard Galerkin approach. When the meshes are quasi-uniform and satisfy the discrete maximum principle, we prove that for a smooth electromagnetic field the crime is of the order of the mesh width

Simplicial gauge theory on spacetime

h

Manifestly gauge invariant discretizations of the Schrödinger equation

, for a Coulomb potential it is of order

Superluminal group velocity in a birefringent crystal

h|\log h|

Discretizing the Maxwell–Klein–Gordon equation by the lattice gauge theory formalism

, and for a general finite energy electromagnetic field it is of order

Convergence of lattice gauge theory for Maxwell’s equations

h^{1/2}

Solving the Maxwell-Klein-Gordon equation in the Lattice Gauge Theory formalism

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Stability of an upwind Petrov Galerkin discretization of convection diffusion equations

We define a discrete gauge-invariant Yang–Mills–Higgs action on spacetime cylindrical meshes with simplicial spatial base. The formulation is a generalization of classical lattice gauge theory, and we prove consistency of the action in the finite element sense. In addition, we perform numerical tests of convergence towards exact continuum results for several choices of gauge fields in pure gauge theory.

Second order gauge invariant discretizations to the Schr\"odinger and Pauli equations

Abstract Grid-based discretizations of the time dependent Schrodinger equation coupled to an external magnetic field are converted to manifest gauge invariant discretizations. This is done using generalizations of ideas used in classical lattice gauge theory, and the process defined is applicable to a large class of discretized differential operators. In particular, popular discretizations such as pseudospectral discretizations using the fast Fourier transform can be transformed to gauge invariant schemes. Also generic gauge invariant versions of generic time integration methods are considered, enabling completely gauge invariant calculations of the time dependent Schrodinger equation. Numerical examples illuminating the differences between a gauge invariant discretization and conventional discretization procedures are also presented.