The fully Sinc-Galerkin method is developed for a family of complex-valued partial differential equations with time-dependent boundary conditions. The Sinc-Galerkin discrete system is formulated and represented by a Kronecker product form of those equations. The numerical solution is efficiently calculated and the method exhibits an exponential convergence rate. Several examples, some with a re...

We consider several extremal problems concerning representations of graphs as distance graphs on the integers. Given a graph G = (V, E), we wish to find an injective function φ : V → Z = {1, 2, . . . } and a set D ⊂ Z such that {u, v} ∈ E if and only if |φ(u)− φ(v)| ∈ D. Let s(n) be the smallest N such that any graph G on n vertices admits a representation (φG,DG) such that φG(v) ≤ N for all v ...

An analysis of a multiscale symmetric interior penalty discontinuous Galerkin finite element method for the numerical discretization of elliptic problems with multiple scales is proposed. This new method, first described in [A. Abdulle, C.R. Acad. Sci. Paris, Ser. I 346 (2008)] is based on numerical homogenization. It allows to significantly reduce the computational cost of a fine scale discont...

We derive energy-norm a posteriori error bounds for an Euler timestepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler timestepping. All results are presented in an abstract setting and then illust...