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Error analysis of a weighted least‐squares finite element method for 2‐D incompressible flows in velocity–stress–pressure formulation
Author(s) -
Yang SuhYuh
Publication year - 1998
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199812)21:18<1637::aid-mma13>3.0.co;2-m
Subject(s) - mathematics , finite element method , mathematical analysis , mixed finite element method , norm (philosophy) , sobolev space , boundary value problem , hilbert space , extended finite element method , physics , political science , law , thermodynamics
In this paper we are concerned with a weighted least‐squares finite element method for approximating the solution of boundary value problems for 2‐D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second‐order problem as a Petrovski‐type first‐order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert‐type boundary conditions. A weighted least‐squares finite element method is proposed for solving this extended first‐order problem. The finite element approximations are defined to be the minimizers of a weighted least‐squares functional over the finite element subspaces of the H 1 product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L 2 ‐norm and in the H 1 ‐norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.