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

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 $\sim 5\times10^3$.