On numerical method for the Stokes problem with Neumann boundary conditions in non-convex domain

A. V. Rukavishnikov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 13 (2024) no. 4, pp. 5-18

Résumé

The Stokes problem with Neumann boundary conditions with an incoming angle at the boundary of the two-dimensional domain is considered. The concept of an $R_{\nu}$-generalized solution in sets of weighted Sobolev spaces is introduced. A weighted finite element method on a uniform grid is constructed based on the second-order Taylor-Hood finite element pair and the introduction to the basis of a weight function in some powers $\nu^{\ast}$ and $\mu^{\ast}$ for the components of the velocity field and the scalar pressure function, respectively. The weight function in the domain coincides with the function of the distance from the point to the vertex of the incoming angle in some $\delta$-neighborhood and the constant $\delta$ outside it. Numerical experiments in the non-convex domain is carried out. The convergence rate of the approximate solution to the exact one is obtained, which is independent of an incoming angle value and exceeds the convergence rate for the classical FEM. The convergence result is achieved without geometric refinement of the mesh in the vicinity of the singularity point. A series of numerical experiments for different values of an incoming angle was carried out and the domain of suitable free parameters of the proposed approach is found. For any point of the constructed domain, an optimal result, from the point of view of convergence, is achieved. The area of choice of suitable free parameters differs from the area for the considered problem with Dirichlet boundary conditions.

Détail

Keywords: angle singularity, Stokes problem with Neumann boundary conditions, $R_{\nu}$-generalized solution, weighted FEM.