. We continue our work on adaptive finite element methods with a study of time discretization of analytic semigroups. We prove optimal a priori and a posteriori error estimates for the discontinuous Galerkin method showing, in particular, that analytic semigroups allow long-time integration without error accumulation. 1. Introduction This paper is a continuation of the series of papers [1], [2], [3], [4], [5] on adaptive finite element methods for parabolic problems. The method considered is the discontinuous Galerkin method (the dG-method) based on a space-time finite element discretization with piecewise polynomial basis functions that are continuous in space and discontinuous in time. In [1], [2], [3], [4], [5] we proved optimal a priori and a posteriori error estimates for the dG-method for parabolic problems, typically of the form: find u : [0; 1) ! H such that u(t) + Au(t) = f(t); t ? 0; u(0) = u 0 ; (1.1) where H = L 2 (\Omega\Gamma with\Omega a bounded domain in R n , Av = ...

Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.

We study four transformations which lead from one well-posed linear system to another: time-inversion, flow-inversion, time-flow-inversion and duality. Time-inversion means reversing the direction of time, flow-inversion means interchanging inputs with outputs, while time-flow-inversion means doing both of the inversions mentioned before. A well-posed linear system is time-invertible if and only if its operator semigroup extends to a group. The system is flow-invertible if and only if its input-output map has a bounded inverse on some (hence, on every) finite time interval [0; ] ( > 0). This is true if and only if the transfer function of has a uniformly bounded inverse on some right half-plane. The system is time-flow-invertible if and only if on some (hence, on every) finite time interval [0; ], the combined operator from the initial state and the input function to the final state and the output function is invertible. This is the case, for example, if the system is conservative, since then is unitary. Time-ow-inversion can sometimes, but not always, be reduced to a combination of time- and flow-inversion. We derive a surprising necessary and sucient condition for to be time-flow-invertible: its system operator must have a uniformly bounded inverse on some left halfplane.

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An overview of various aspects related to the spectral and nonlinear stability of travelling-wave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green's functions) for the linear operator. Among the other topics reviewed in this survey are the nonlinear stability of waves, the stability and interaction of wellseparated multi-bump pulses, the numerical computation of spectra, and the Evans function, which is a tool to locate isolated eigenvalues in the point spectrum and near the essential spectrum. Furthermore, methods for the stability of waves in Hamiltonian and monotone equations as well as for singularly perturbed problems are mentioned. Modulated waves, rotating waves on the plane, and travelling waves on cylindrical domains are also discussed briefly.

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diusion model is proven.

An equivariant center-manifold reduction near relative equilibria of G-equivariant semiflows on Banach spaces is presented. In contrast to previous results, the Lie group G is possibly non-compact. Moreover, it is not required that G induces a strongly continuous group action on the underlying function space. In fact, G may act discontinuously. The results are applied to bifurcations of stable patterns arising in reactiondiffusion systems on the plane or in three-space modeling chemical systems such as catalysis on platinum surfaces and Belousov-Zhabotinsky reactions. These systems are equivariant under the Euclidean symmetry group. Hopf bifurcations from rigidlyrotating spiral waves to meandering or drifting waves, and from twisted scroll rings are investigated.

. A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate. 1. Introduction. The Cahn-Hilliard equation (1.1) u t + \Delta 2 u \Gamma \DeltaOE(u) = 0; x 2\Omega ae R 3 ; t ? 0; w...

Abstract. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropicdiffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak–Keller–Segel–Alt model for chemotaxis.

In applications, solitary-wave solutions of semilinear elliptic equations \Deltau + g(u; ru) = 0 (x; y) 2 IR \Theta\Omega in infinite cylinders frequently arise as travelling waves of parabolic equations. As such, their bifurcations are an interesting issue. Interpreting elliptic equations on infinite cylinders as dynamical systems in x has proved very useful. Still, there are major obstacles in obtaining, for instance, bifurcation results similar to those for ordinary differential equations. In this article, persistence and continuation of exponential dichotomies for linear elliptic equations is proved. With this technique at hands, Lyapunov-Schmidt reduction near solitary waves can be applied. As an example, existence of shift dynamics near solitary waves is shown if a perturbation ¯h(x; u; ru) periodic in x is added.