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

Author:
Ching Lung Chang

Journal:
Math. Comp. **63** (1994), 41-50

MSC:
Primary 65N15; Secondary 65N30, 76D07, 76M10

DOI:
https://doi.org/10.1090/S0025-5718-1994-1234425-7

MathSciNet review:
1234425

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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 and pressure *p* as unknowns. With the additional variable , the second-order problem is reduced to a first-order system. Considering the compatibility condition , 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 , and *p*, this method achieves optimal rates of convergence in the -norms.

**[1]**S. Agmon, A. Douglis, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying boundary conditions*. II, Comm. Pure Appl. Math.**17**(1964), 35-92. MR**0162050 (28:5252)****[2]**A. K. Aziz, R. B. Kellogg, and A. B. Stephens,*Least squares methods for elliptic systems*, Math. Comp.**44**(1985), 53-70. MR**771030 (86i:65069)****[3]**A. K. Aziz and J. L. Liu,*A weighted least squares method for the backward-forward heat equation*, SIAM J. Numer. Anal.**28**(1991), 156-167. MR**1083329 (92a:65333)****[4]**I. Babuška,*The finite element method with Lagrangian multipliers*, Numer. Math.**20**(1973), 179-192. MR**0359352 (50:11806)****[5]**I. Babuška, J. T. Oden, and K. Lee,*Mixed-hybrid finite element approximations of second-order boundary value problems*, Comput. Methods Appl. Mech. Engrg.**11**(1977), 175-206. MR**0451771 (56:10053)****[6]**J. H. Bramble and R. Scott,*Simultaneous approximation in scales of Banach spaces*, Math. Comp.**32**(1978), 947-954. MR**501990 (80a:65222)****[7]**J. H. Bramble and A. H. Schatz,*Least squares for 2mth order elliptic boundary-value problems*, Math. Comp.**25**(1971), 1-32. MR**0295591 (45:4657)****[8]**F. Brezzi,*On the existence, uniqueness, and approximation of saddle-point problems arising from Lagrange multipliers*, RAIRO Anal. Numer.**8**(1974), 129-151. MR**0365287 (51:1540)****[9]**F. Brezzi and J. Douglas, Jr.,*Stabilized mixed methods for the Stokes problem*, Numer. Math.**53**(1988), 225-235. MR**946377 (89g:65138)****[10]**G. F. Carey and B. N. Jiang,*Least-squares finite elements for first-order hyperbolic systems*, Internat. J. Numer. Mech. Engrg.**26**(1988), 81-93. MR**921572 (88k:65092)****[11]**C.-L. Chang,*A finite element method for first order elliptic systems of*3-D, Appl. Math. Comput.**23**(1987), 171-184. MR**896976 (89m:65100)****[12]**C.-L. Chang and M. D. Gunzburger,*A subdomain Galerkin/least squares method for first-order elliptic systems in the plane*, SIAM J. Numer. Anal.**27**(1990), 1197-1221. MR**1061126 (91i:65176)****[13]**C.-L. Chang and B. N. Jiang,*An error analysis of least-squares finite element method of velocity-pressure-vorticity formulation for Stokes problem*, Comput. Methods Appl. Mech. Engrg.**84**(1990), 247-255. MR**1082823 (91k:76106)****[14]**P. Ciarlet,*The finite element method for elliptic problems*, North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[15]**G. J. Fix, M. D. Gunzburger, and R. A. Nicolaides,*On finite element methods of the least squares type*, Comput. Math. Appl.**5**(1979), 87-98. MR**539567 (81b:65103)****[16]**G. J. Fix and M. E. Rose,*A comparative study of finite element and finite difference methods for Cauchy-Riemann type equations*, SIAM J. Numer. Anal.**22**(1985), 250-261. MR**781319 (86g:65193)****[17]**V. Girault and P. A. Raviart,*Finite element methods for Navier-Stokes equations*, Springer-Verlag, Berlin, 1986. MR**851383 (88b:65129)****[18]**B. N. Jiang and C.-L. Chang,*Least-squares finite elements for Stokes problem*, Comput. Methods Appl. Mech. Engrg.**78**(1990), 297-311. MR**1039687 (91h:76058)****[19]**B. N. Jiang and L. A. Povinelli,*Least-squares finite element method for fluid dynamics*, Comput. Methods Appl. Mech. Engrg.**81**(1990), 13-37. MR**1071091 (91f:76040)****[20]**C. Miranda,*Partial differential equations of elliptic type*, 2nd rev. ed. (Zane C. Motteler, translator), Springer-Verlag, Berlin, 1970. MR**0284700 (44:1924)****[21]**P. Neittaamäki and J. Saranen,*Finite element approximation of vector fields given by curl and divergence*, Math. Methods Appl. Sci.**3**(1981), 328-335. MR**657301 (83e:65193)****[22]**R. Temam,*Navier-Stokes equations and nonlinear functional analysis*, SIAM, Philadelphia, PA, 1983. MR**764933 (86f:35152)****[23]**W. L. Wendland,*Elliptic systems in the plane*, Pitman, London, 1979. MR**518816 (80h:35053)****[24]**O. C. Zienkiewicz,*The finite element method*, Vol. 1, 4th ed., McGraw-Hill, New York, 1989.**[25]**O. C. Zienkiewicz, D. R. J. Owen, and K. Niles,*Least-squares finite element for elasto-static problems--use of reduced integration*, Internat. J. Numer. Methods Engrg.**8**(1974), 341-358.

Retrieve articles in *Mathematics of Computation*
with MSC:
65N15,
65N30,
76D07,
76M10

Retrieve articles in all journals with MSC: 65N15, 65N30, 76D07, 76M10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1994-1234425-7

Article copyright:
© Copyright 1994
American Mathematical Society