Dietrich Braess

Über ein Paradoxon aus der Verkehrsplanung

ZusammenfassungFür die Straßenverkehrsplanung möchte man den Verkehrsfluß auf den einzelnen Straßen des Netzes abschätzen, wenn die Zahl der Fahrzeuge bekannt ist, die zwischen den einzelnen Punkten des Straßennetzes verkehren. Welche Wege am günstigsten sind, hängt nun nicht nur von der Beschaffenheit der Straße ab, sondern auch von der Verkehrsdichte. Es ergeben sich nicht immer optimale Fahrzeiten, wenn jeder Fahrer nur für sich den günstigsten Weg heraussucht. In einigen Fällen kann sich durch Erweiterung des Netzes der Verkehrsfluß sogar so umlagern, daß größere Fahrzeiten erforderlich werden.SummaryFor each point of a road network let be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of the traffic flow. Whether a street is preferable to another one depends not only upon the quality of the road but also upon the density of the flow. If every driver takes that path which looks most favorable to him, the resultant running times need not be minimal. Furthermore it is indicated by an example that an extension of the road network may cause a redistribution of the traffic which results in longer individual running times.

On a Paradox of Traffic Planning

For each point of a road network, let there be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times.

A New Convergence Proof for the Multigrid Method Including the V-Cycle

For a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem. Hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the “V-cycle”. The presented proof applies to procedures with any number of smoothing iterations and to the V-cycle.

Towards algebraic multigrid for elliptic problems of second order

An algebraic multigrid method is developed which can be used as a preconditioner for the solution of linear systems of equations with postitive definite matrices. The method is directed to equations which arise from the discretization of elliptic equations of second order, but only the matrix is the source for the information used by the algorithm. One has only to know whether the matrix stems from a 2-dimensional or 3-dimensional problem and whether the elliptic equations are scalar equations or belong to a system.ZusammenfassungEs wird ein algebraisches Mehrgitterverfahren vorgestellt, das zur Vorkonditionierung von positiv definiten Matrizen geeignet ist. Es wurde entworfen für Gleichungssysteme, die aus der Diskretisierung von elliptischen Differentialgleichungen stammen. Alle Information wird aus der Matrix herausgezogen. Man braucht nur zu wissen, ob ein zwei- oder dreidimensionales Problem und ob eine skalare Gleichung oder ein System zugrunde liegt.

A Posteriori Error Estimators for the Raviart--Thomas Element

When error estimators for the \RTe\ are developed, two difficulties prevent the success of the straightforward application of frequently used arguments. The

An efficient smoother for the Stokes problem

\Hdiv

Equilibrated residual error estimator for edge elements

-norm is an anisotropic norm; i.e., it refers to differential operators of different orders. Moreover, the traces of

A Multigrid Algorithm for the Mortar Finite Element Method

\Hdiv

The contraction number of a multigrid method for solving the Poisson equation

-functions are only in

A posteriori error estimators for obstacle problems – another look

H^{-1/2}