In the present paper a method by Lehmann-Maehly and Goerisch is extended to self-adjoint eigenvalue problems with arbitrary essential spectrum. This extension is obtained by consequently making use of the local character of the method. In this way, upper and lower bounds to all isolated eigenvalues are derived. In our proofs, the close relationship to Wielandt's inverse iteration becomes quite obvious.

The Method of Transport is a genuinely multi-dimensional scheme to solve nonlinear systems of hyperbolic equations numerically. It is based on the framework of conservation laws. Here, we will consider the Euler equations. We will present an alternative formulation of the rst order method based on kinetic theory. This will allow us to show that density and pressure of the numerical solution remain positive for all times. In addition, we can derive L 1 -estimates for the numerical solution. We will also consider the second order method. This will give us more insight into the di erences of the two formulations.

Abstract.In this paper, we present some interesting connections between a number of Riemannsolver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey's Method of Transport (MoT) (respectively the second author's ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach. Furthermore, Ostkamp's equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach. At the same time, we clarify some important differences between these two approaches.Mathematics Subject Classification. 35C15, 35L65, 65D32, 65M25, 76M12, 76N15, 76P05.

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