Stokes Problem, Layer Potentials and Regularizations, and Multiscale Applications

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).