Second-order accurate (up to the axis of symmetry) finite-element implementations of iterative methods with splitting of boundary conditions for Stokes and stokes-type systems in a spherical layer

Previously, numerical implementations of methods with splitting of boundary conditions for solving the first boundary value problem for Stokes and Stokes-type systems in a spherical layer with axial symmetry were developed on the basis of bilinear finite elements in a spherical coordinate system. These finite-element implementations are second-order accurate outside the neighborhood of the axis of symmetry, but their accuracy reduces near the axis of symmetry (down to the first order for pressure). Recently, new linear-type second-order accurate (up to the poles) finite-element approximations of the Laplace–Beltrami operators and the angular components of the gradient and divergence operators on a sphere in $\mathbb R^3$ in the axisymmetric case, as well as corresponding finite-element spaces, have been found by the authors. These finite-element approximations and spaces are used here to modify the above finite-element implementations of methods with splitting of boundary conditions for Stokes and Stokes-type systems. The finite-element schemes arising at iterations are written using one-dimensional tridiagonal operators with respect to angular and radial variables, which makes it possible to accelerate the computations nearly twofold. Numerical experiments reveal that the modified finite-element implementations of methods are second-order accurate with respect to the mesh size in the max-norm over the entire spherical layer. The new numerical method for the Stokes system is highly accurate with respect to both velocity and pressure. At the same time, in actual cases arising in implicit time discretizations of an initial-boundary value problem for the nonstationary Stokes system, when the singular parameter is large and the time step $\tau$ is small, the numerical methods constructed for Stokes-type systems become highly inaccurate with respect to pressure, while preserving sufficient accuracy of velocity. It is shown that sufficiently high accuracy of both velocity and pressure can be achieved under the condition $\tau\sim h$, where $h$ is the characteristic mesh size of the spatial grid. A numerical experiment is described that shows how the accuracy of numerical solutions can be considerably improved for such Stokes-type systems occurring in reality.