# IJPAM: Volume 37, No. 3 (2007)

**APPROXIMATE APPROXIMATIONS AND A BOUNDARY**

POINT METHOD FOR THE LINEARIZED STOKES SYSTEM

POINT METHOD FOR THE LINEARIZED STOKES SYSTEM

Faculty of Mathematics, FB 17

University of Kassel

Kassel, D-34109, GERMANY

e-mail: sergejk@mathematik.uni-kassel.de

e-mail: varnhorn@mathematik.uni-kassel.de

**Abstract.**The method of approximate approximations,
introduced by Maz'ya [#!mazya91!#], can also be used for the
numerical solution of boundary integral equations. In this case,
the matrix of the resulting algebraic system to compute an
approximate source density depends only on the position of a
finite number of boundary points and on the direction of the
normal vector in these points (Boundary Point Method). We
investigate this approach for the Stokes problem in the whole
space and for the Stokes boundary value problem in a bounded
convex domain , where the second part consists of
three steps: In a first step the unknown potential density is
replaced by a linear combination of exponentially decreasing basis
functions concentrated near the boundary points. In a second step,
integration over the boundary is replaced by
integration over the tangents at the boundary points such that
even analytical expressions for the potential approximations can
be obtained. In a third step, finally, the linear algebraic system
is solved to determine an approximate density function and the
resulting solution of the Stokes boundary value problem. Even not
convergent the method leads to an efficient approximation of the
form
, where can be chosen
arbitrarily small.

**Received: **May 14, 2007

**AMS Subject Classification: **31B10, 35J05, 41A30, 65N12, 76D07

**Key Words and Phrases: **approximate approximations, boundary point method, Stokes potentials

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2007

**Volume:** 37

**Issue:** 3