An error estimate of the least squares finite element method for the Stokes problem in three dimensions

In this paper we are concerned with the Stokes problem in three dimensions (see recent works of the author and B. N. Jiang for the two-dimensional case). It is a linear system of four PDEs with velocity 11 and pressure p as unknowns. With the additional variable a = curl&, the second-order problem is reduced to a first-order system. Considering the compatibility condition d i v g = 0 , we have a system with eight first-order equations and seven unknowns. A least squares method is applied to this extended system, and also to the corresponding boundary conditions. The analysis based on works of Agmon, Douglis, and Nirenberg; Wendland; Zienkiewicz, Owen, and Niles; etc. shows that this method is stable in the h-version. For instance, if we choose continuous piecewise polynomials to approximate 11, a,and p , this method achieves optimal rates of convergence in the HI-norms. Let SZ be an open bounded and connected subset of IR3 with a smooth boundary T. Let be a given function representing the body f E [ L ~ ( R ) ] ~ force. The Stokes can be posed as where g , p with ( p , 1) = 0 , and v are respectively velocity, pressure, and kinematic viscosity (constant), all of which are assumed to be nondimensionalized. Over the past two decades many engineers and mathematicians have studied the above problem. The mixed Galerkin method solves this problem successfully. In most cases the elements are required to satisfy a saddle point condition [4, 5, 8, 9, 221, which is not necessary for our method. Received by the editor March 16, 1992 and, in revised form, October 12, 1992. 1991 Mathematics Subject Classification. Primary 65N30, 35F15. This work was performed when the author was an OAIJCWRU Summer Faculty fellow participant at NASA Lewis Research Center, Cleveland, OH (1991), and was revised at the University of Texas at Arlington (1992) and at Wright-Patterson Air Force Base, Dayton, OH (1993). @ 1994 American Mathematical Society 0025-5718194 $1.00 + $.25 per page

[1]  Graham F. Carey,et al.  Least‐squares finite elements for first‐order hyperbolic systems , 1988 .

[2]  R. B. Kellogg,et al.  Least Squares Methods for Elliptic Systems , 1985 .

[3]  W. Wendland.  Elliptic systems in the plane , 1979 .

[4]  J. Z. Zhu,et al.  The finite element method , 1977 .

[5]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[6]  Max Gunzburger,et al.  A subdomain Galerkin/Least squares method for first-order elliptic systems in the plane , 1990 .

[7]  James H. Bramble,et al.  Least squares methods for 2th order elliptic boundary-value problems , 1971 .

[8]  James H. Bramble,et al.  Least Squares Methods for 2mth Order Elliptic Boundary-Value Problems , 1971 .

[9]  A. Aziz,et al.  A weighted least squares method for the backward-forward heat equation , 1991 .

[10]  B. Jiang,et al.  Least-square finite elements for Stokes problem , 1990 .

[11]  B. Jiang,et al.  Least-squares finite element method for fluid dynamics , 1990 .

[12]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[13]  Max Gunzburger,et al.  A finite element method for first order elliptic systems in three dimensions , 1987 .

[14]  Pekka Neittaanmäki,et al.  Finite element approximation of vector fields given by curl and divergence , 1981 .

[15]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[16]  O. C. Zienkiewicz,et al.  Least square-finite element for elasto-static problems. Use of `reduced' integration , 1974 .

[17]  L. R. Scott,et al.  Simultaneous approximation in scales of Banach spaces , 1978 .

[18]  J. Douglas,et al.  Stabilized mixed methods for the Stokes problem , 1988 .

[19]  I. Babuska,et al.  Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems. Interim report , 1975 .

[20]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[21]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[22]  Ching L. Chang,et al.  An error analysis of least-squares finite element method of velocity-pressure-vorticity formulation for Stokes problem , 1990 .

[23]  Max Gunzburger,et al.  On finite element methods of the least squares type , 1979 .

[24]  Milton E. Rose,et al.  A COMPARATIVE STUDY OF FINITE ELEMENT AND FINITE DIFFERENCE METHODS FOR CAUCHY-RIEMANN TYPE EQUATIONS* , 1985 .