New approximate method for solving the Stokes problem in a domain with corner singularity

In this paper we introduce the notion of an $R_{\nu}$-generalized solution to the Stokes problem with singularity in a two-dimensional non-convex polygonal domain with one reentrant corner on its boundary in special weight sets. We construct a new approximate solution of the problem produced by weighted finite element method. An iterative process for solving the resulting system of linear algebraic equations with a block preconditioning of its matrix is proposed on the basis of the incomplete Uzawa algorithm and the generalized minimal residual method. Results of numerical experiments have shown that the convergence rate of the approximate $R_{\nu}$-generalized solution to an exact one is independent of the size of the reentrant corner on the boundary of the domain and equals to the first degree of the grid size $h$ in the norm of the weight space $W^1_{2,\nu}(\Omega)$ for the velocity field components in contrast to the approximate solution produced by classical finite element or finite difference schemes convergence to a generalized one no faster than at an $\mathcal{ O}(h^{\alpha})$ rate in the norm of the space $W^1_{2}(\Omega)$ for the velocity field components, where $\alpha<1$ and $\alpha$ depends on the size of the reentrant corner.