The numerical integration of many geometric partial differential equations involves discrete approximations of several firstand second-order geometric differential operators. In this paper, we consider consistent discretized approximations of these operators based on a quadratic fitting scheme. An asymptotic error analysis is conducted which shows that the discrete approximations of the firstan...

led to a theory relating periodic second order (differential and difference) operators to hyperelliptic curves with branch points given by the periodic and antiperiodic spectrum of the original operator. As a result the periodic second order operators with a given spectrum form a torus (except for a lower dimensional submanifold) which is the Jacobi variety of the defining curve. Krichever [15,...

We construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are Lp stable for any real number p ∈ [1,∞], and commute with the differential operators ∇, ∇×, and ∇·. We also construct mollification operators satisfying boundary conditions and use them to characterize the kernel of traces related to the tangential and normal trace of vector fields....

In this work, we construct commutative rings of two variable matrix differential operators that are isomorphic to a ring of meromorphic functions on a rational manifold obtained from the CP 1×CP 1 by identification of two lines with the pole on a certain rational curve. The commutation condition for differential operators is equivalent to a system of non-linear equations in the operators’ coeff...

A model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Preprint 2013]. The proof relies on a new lifting lemma in differential algebra: a diff...

The two papers in this series analyze quantum invariant differential operators for quantum symmetric spaces in the maximally split case. In this paper, we complete the proof of a quantum version of Harish-Chandra’s theorem: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and a ring of Laurent polynomial invariants with resp...