Abstract:
In this study the iteration of boundary conditions method (Chizhonkov and Kargin, 2006) is
used together with the well known Finite difference numerical method to solve the Stokes
problem over a rectangular domain as well as in irregular domain. The iteration of boundary
conditions method has been applied to the Stokes problem in a rectangular domain,
2
<x<
2
,
d
2
< y <
d
2
, by the above mentioned researchers. Our main task here is
to validate the results of the approximate methods by this analytical method in case of the
rectangular domain and extend that to the case of irregular domain.The (Chizhonkov and
Kargin, 2006) algorithm is typically the best choice for validation purposes because of its
high accuracy.
It is known in literature that increasing the parameter d, which represents the ratio of the
sides, leads to slow down in convergence of the approximate methods like the conjugate
Gradients of Uzawa (Kobelkov and Olshanskii, 2000). It is therefore important that an
algorithm that converges uniformly with respect to the parameter d is considered. The
(Chizhonkov and Kargin, 2006) algorithm is typical of such an algorithm, and hence our
choice of the method in this work.
In this project the non-homogeneous Stokes problem is transformed into a homogeneous
Stokes problem and the resulting problem is then decomposed into two sub problems that
are solvable by the eigenfunction expansion method. Once all necessary coefficients of the
generalised Fourier series are known and the functions describing the boundary conditions
are prescribed and represented in terms of the Fourier series, we then proceed to formulate
the iteration of boundary conditions numerical algorithm. Finally we develop a numerical
scheme, using the finite difference methods, for solving the problem in both rectangular and
irregular domains. Coding of the numerical algorithm is done using MATLAB 9.0,R2016a
programming language, and implemented by the author. The results of the two methods in
both cases of boundary conditions are then compared for validation of our purely numerical
results.
