Stokes Problem, Layer Potentials and Regularizations, and Multiscale Applications

Carsten Mayer () and Willi Freeden ()
Additional contact information
Carsten Mayer: University of Kaiserslautern, Geomathematics Group
Willi Freeden: University of Kaiserslautern, Geomathematics Group

A chapter in Handbook of Geomathematics, 2015, pp 1155-1253 from Springer

Abstract: Abstract This Layer Potentials contribution represents a continuation of a series of papers by the authors concerned with the multiscale solution of boundary-value problems corresponding to elliptic differential equations such as Laplace equation (Freeden and Mayer, Appl Comput Harmonic Anal 14:195–237, 2003; Acta Geod Geophys Hung 41:55–86, 2006), Helmholtz equation (Freeden et al., Numer Funct Anal Optim 24:747–782, 2003; Ilyasov, A tree algorithm for Helmholtz potential wavelets on non-smooth surfaces: theoretical background and application to seismic data postprocessing. PhD thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2011), Cauchy-Navier equation (Abeyratne, Cauchy-Navier wavelet solvers and their application in deformation analysis. PhD thesis, Geomathematics Group, University of Kaiserslautern, 2003; Abeyratne et al., J Appl Math 12:605–645, 2003), and Maxwell equations (Freeden and Mayer, Int J Wavelets Multiresolut Inf Process 5:417–449, 2007). The essential idea is to transform the differential equation into an integral equation by standard surface layer potentials and to use certain regularizations of the kernels of the layer potentials as scaling kernel functions. In this context, the distance of a parallel surface to the boundary acts as the scale parameter. The scaling kernel functions are defined as restrictions of the kernel values of layer potentials to the parallel surface. Wavelet kernel functions in scale discrete case are canonically obtained as the difference between two consecutive scaling functions. The solution process is formulated in such a way that an approximation of a boundary function on a regular surface simultaneously yields the solution of the boundary-value problem itself. In the case of Stokes flow – as discussed here – the kernels are of vectorial/tensorial nature, satisfying the differential equation in each variable. Stokes flow leads to significant applications of geomathematical relevance (e.g., in oceanography, meteorology).

Keywords: Double Layer Potential; Regular Surface; Scaling Function; Abeyratne; Exterior Stokes Problem (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-54551-1_95

Ordering information: This item can be ordered from
http://www.springer.com/9783642545511

DOI: 10.1007/978-3-642-54551-1_95

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().