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Integral equation methods for problems of mathematical physics

Collaborator: G. Schmidt

Cooperation with: V. Maz'ya, (Linköping University, Sweden), W. Wendland (Universitšt Stuttgart), V. Karlin, (University of Lancashire, Preston, UK)

Supported by: (EU) INTAS: ``Development of constructive and numerical methods for solving nonlocal linear and nonlinear problems for partial differential equations''

Description: The project ``Development of constructive and numerical methods for solving nonlocal linear and nonlinear problems for partial differential equations'', which was coordinated by the Weierstrass Institute, was supported by INTAS from 10/1999 till 10/2002. Groups from Germany, Sweden, Italy, Russia, Belorussia, Georgia, and Ukraine participated in this project.

Within the project in 2002 the following topics were studied:

In [1] a new cubature method for surface potentials via approximate approximations was developed. It is applied to the computation of multi-dimensional single layer harmonic potentials. Due to singularities under the integral sign, in the multivariate case usual cubature methods are very expensive if the potential has to be computed close to the surface.

Our approach uses the asymptotic expansion of surface integrals by integrals over the tangential plane, which can be transformed to one-dimensional ones for radial basis functions and provides therefore efficient high order cubature formulas for surface potentials.

In [2] we developed and implemented a new numerical method for solving the Sivashinsky equation, which models dynamics of the long wave flame instability. The discretization of this nonlocal nonlinear evolution equation is based on approximate approximations, which allows the exact computation of singular integrals.

Extended numerical experiments confirm theoretical results on the convergence of the method. They are used to study the sensitivity of flame fronts with respect to external perturbations.

References:

  1. V. MAZ'YA, G. SCHMIDT, W. WENDLAND, On the computation of multi-dimensional single layer harmonic potentials via approximate approximations, to appear in: Math. Models Methods Appl. Sci.
  2. V. KARLIN, V. MAZ'YA, G. SCHMIDT, Numerical algorithms to calculate periodic solutions of the Sivashinsky equation, WIAS Preprint no. 771, 2002.



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5/16/2003