We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and parameterized nonlinear wave equations. The aim of the paper is to present and analyze a continuation method for invariant subspaces that belong to a group of eigenvalues, the number of which is much smaller than the dimension of the system. The continuation method is of predictor-corrector type, similar to the approach for the linear eigenvalue problem in [Beyn, Kleß, and Thümmler, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001], but we avoid linearizing the problem, which will double the dimension and change the sparsity pattern. The matrix equations that occur in the predictor and the corrector step are solved by a bordered version of the Bartels-Stewart algorithm. Furthermore, we set up an update procedure that handles the transition from real to complex conjugate eigenvalues, which occurs when eigenvalues from inside the continued cluster collide with eigenvalues from outside. The method is demonstrated on several numerical examples: a homotopy between random matrices, a fluid conveying pipe problem, and a traveling wave of a damped wave equation.
The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a comoving frame in which the solution becomes stationary. In addition it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type.Key words. Systems of damped wave equations, traveling waves, nonlinear stability, freezing method, second order evolution equations, point spectra and essential spectra.
We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete solution sets. Preliminaries from set-valued analysisWe refer to the monographs [1] and [18] for general notions from setvalued analysis. In the following we specify some notation that will be used throughout this paper. In the following, let X and Y be normed spaces.Definition 1. For any x ∈ X and any subset M ⊆ X, we define the distance of x to M by dist(x, M ) X = inf{ x − y X : y ∈ M } and the proximal set byRecall that proj(x, M ) X is a singleton in case M is convex and X is a Hilbert space. Moreover, by a common abuse of notation, we write M X := sup x∈M x X for M ⊆ X bounded.By CBC(X) we denote the set of all closed, bounded, and convex subsets of X. There are various ways of defining a topology on CBC(X) which in general are not equivalent. We will use convergence in the Hausdorff and in the Kuratowski sense. Definition 2. For any two sets M, M ⊆ X, the Hausdorff semidistance and the Hausdorff distance are defined by dist(M, M ) X := sup x∈M dist(x, M ) X , dist H (M, M ) X := max{dist(M, M ) X , dist( M , M ) X }.It is well-known that dist H defines a metric on CBC(X).
Implicit function theorems are derived for nonlinear set valued equations that satisfy a relaxed one-sided Lipschitz condition. We discuss a local and a global version and study in detail the continuity properties of the implicit set-valued function. Applications are provided to the Crank-Nicolson scheme for differential inclusions and to the analysis of differential algebraic inclusions.
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