A staggered cell‐centered finite element method for Stokes problems with variable viscosity on general meshes

• Published in: Numerical methods for partial differential equations Vol. 39; no. 2; pp. 1729 - 1766
• Main Authors: Huu Du, Nguyen, Ong, Thanh Hai
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.03.2023; Wiley Subscription Services, Inc
• Subjects: Approximation; cell‐centered schemes; Continuity (mathematics); error estimates; Finite element method; finite elements; Linear functions; Mathematical analysis; Stokes equations; Viscosity
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.22952

In this paper, we propose to extend the staggered cell‐centered finite element method (SCFEM) on general meshes for the Stokes problems with variable viscosity (possibly discontinuous). The scheme is cell‐centered in the sense that the solution can be computed by cell unknowns of the primal mesh and the dual mesh for the velocity and the pressure, respectively, where the velocity is approximated by piecewise linear functions (ℙ1$$ {\mathrm{\mathbb{P}}}^1 $$) on the triangular dual submesh, and the pressure is approximated by piecewise constant functions (ℙ0$$ {\mathrm{\mathbb{P}}}^0 $$) on the dual mesh. In order to get the local continuity of numerical stresses across the interfaces, the scheme gives the auxiliary edge unknowns interpolated by the multipoint stress approximation technique. Its stability and convergence properties are presented in the rigorous theoretical framework. Numerical results are carried out to highlight accuracy and computational cost.