Stable and unstable cross-grid PkQl mixed finite elements for the Stokes problem

Authors:
María G. Armentano;Jordi Blasco
Affiliations:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina;Departament de Matemítica Aplicada I, Universitat Politècnica de Catalunya, Campus Sud, Edifici H, Avgda. Diagonal 647, 08028, Barcelona, Spain
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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Abstract

In this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the Stokes problem in two space dimensions. In these schemes, the pressure is interpolated on a mesh of rectangular elements, while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals. Continuous interpolations of degrees k for the velocity and l for the pressure are considered, so the new finite elements are called cross-grid P"kQ"l. A stability analysis of these approximations is provided, based on the macroelement technique of Stenberg. The lowest order P"1Q"1 and P"2Q"1 cases are analyzed in detail; in the first case, a global spurious pressure mode is shown to exist, so this element is unstable. In the second case, however, stability is rigorously proved. Numerical results obtained in these two cases are also presented, which confirm the existence of the spurious pressure mode for the P"1Q"1 element and the stability of the P"2Q"1 element.