Heterogenous multiscale method for locally self-similar problems

Multiscale methods for wave problems in heterogenous media

In this paper we give a survey on various multiscale methods for the numerical solution of second order hyperbolic equations in highly heterogenous media. We concentrate on the wave equation and distinguish between two classes of applications. First we discuss numerical methods for the wave equation in heterogeneous media without scale separation. Such a setting is for instance encountered in t...

Kinematic self-similar locally rotationally symmetric models

A brief summary of results on kinematic self-similarities in general relativity is given. Attention is focussed on locally rotationally symmetric models admitting kinematic self-similar vectors. Coordinate expressions for the metric and the kinematic self-similar vector are provided. Einstein’s field equations for perfect fluid models are investigated and all the homothetic perfect fluid soluti...

Numerical method for elliptic multiscale problems

Multiscale representations: fractals, self-similar random processes and wavelets

2 Principles 4 2.1 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Definition and history . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Hölder exponent and singularity spectrum . . . . . . . . . . . . . 6 2.2 Self-similar random processes . . . . . . . . . . . . . . ....

Self-similar power transforms in extrapolation problems

A method is suggested allowing for the improvement of accuracy of self-similar factor and root approximants, constructed from asymptotic series. The method is based on performing a power transform of the given asymptotic series, with the power of this transformation being a control function. The latter is defined by a fixed-point condition, which improves the convergence of the sequence of the ...