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On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers
B. V. Pal'tsev,
I. I. Chechel' Dorodnicyn Computing Center, Russian Academy of Sciences,
ul. Vavilova 40, Moscow, 119991, Russia
Abstract:
The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For
$R/r$ exceeding about 30, where
$r$ and
$R$ are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large
$R/r$) than the convergence rate of its differential version. For this
reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have
revealed that this modification has the same convergence rate as its differential counterpart for
$R/r$ of up to
$5\times10^3$. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from
$R/r\sim30$ and is considerably more efficient for large values of
$R/r$. It is also established that the convergence rates of both methods depend little on the stretching coefficient
$\eta$ of circularly rectangular mesh cells in a range of
$\eta$ that is well sufficient for effective use of the multigrid method for arbitrary values of
$R/r$ smaller than
$\sim 5\times10^3$.
Key words:
stationary Stokes system, spherical gaps, iterative methods with splitting of boundary conditions, second-order accurate finite-element implementations in the axisymmetric case, convergence rates, multigrid method.
UDC:
519.634 Received: 02.12.2005
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