We prove optimal convergence rates for the approximation provided by the original discontinuous Galerkin method for the transport-reaction problem. This is achieved in any dimension on meshes related in a suitable way to the possibly variable velocity carrying out the transport. Thus, if the method uses polynomials of degree k, the L2-norm of the error is of order k+1. Moreover, we also show th...

This paper is concerned with numerical solutions of large scale Sylvester equations AX −XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li andWhite (2002) demonstrated that the so called Cholesky factored ADI method with decent shift parameters can be very effective. In this paper we present a ge...

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of “finite element Hessian” and a Schur complement approach to solvin...

We consider the question of approximating the inverse W = V of the Galerkin stiffness matrix V obtained by discretizing the simple-layer operator V with piecewise constant functions. The block partitioning of W is assumed to satisfy any of the standard admissibility criteria that are employed in connection with clustering algorithms to approximate the discrete BEM operator V. We show that W can...

Slow Invariant Manifolds (SIM) are calculated for isothermal closed reaction-diffusion systems as a model reduction technique. Diffusion effects are examined using a Galerkin projection that rigorously accounts for the coupling of reaction and diffusion processes. This method reduces the infinite dimensional dynamical system by projecting it on a low dimensional approximate inertial manifold. A...

In this paper we study the so-called spin-boson system, namely a two-level system in interaction with a distinguished mode of a quantized bosonic field. We give a brief description of the controlled Rabi and Jaynes–Cummings models and we discuss their appearance in the mathematics and physics literature. We then study the controllability of the Rabi model when the control is an external field a...

The purpose of this note is to study the Euclidean condition number of the matrix resulting from using the well-known RayleighRitz-Galerkin method with finite dimensional subspaces of polynomial spline functions to approximate the solution of a linear, selfadjoint* two-point boundary value problem. Roughly speaking, we consider a model class of such problems of order In and determine upper boun...

In this paper we prove an approximate controllability result for the bilinear Schrödinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof ...

This paper extends for the first time Schaback’s linear discretization theory to nonlinear operator equations, relying heavily on the methods in Böhmer’s 2010 book. There is no restriction to elliptic problems or to symmetric numerical methods like Galerkin techniques. Trial spaces can be arbitrary, including spectral and meshless methods, but have to approximate the solution well, and testing ...

In this paper, we bring three theorems that enable us to approximate the solution of Ferdholm integral equations of the second kind. Then we use the Coifman wavelets or Coiflets as scaling functions for projection that satisfied the conditions of theorems for approximation. Also we use this projection to convert the integral equation to a Galerkin system, which is the most important of the expa...