Analysis of least squares finite element methods for the Stokes equations
Authors:
Pavel B. Bochev and Max D. Gunzburger
Journal:
Math. Comp. 63 (1994), 479506
MSC:
Primary 76M10; Secondary 65N30, 76D07
MathSciNet review:
1257573
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Abstract: In this paper we consider the application of least squares principles to the approximate solution of the Stokes equations cast into a firstorder velocityvorticitypressure system. Among the most attractive features of the resulting methods are that the choice of approximating spaces is not subject to the LBB condition and a single continuous piecewise polynomial space can be used for the approximation of all unknowns, that the resulting discretized problems involve only symmetric, positive definite systems of algebraic equations, that no artificial boundary conditions for the vorticity need be devised, and that accurate approximations are obtained for all variables, including the vorticity. Here we study two classes of least squares methods for the velocityvorticitypressure equations. The first one uses norms prescribed by the a priori estimates of Agmon, Douglis, and Nirenberg and can be analyzed in a completely standard manner. However, conforming discretizations of these methods require continuity of the finite element spaces, thus negating the advantages of the velocityvorticitypressure formulation. The second class uses weighted norms of the residuals to circumvent this flaw. For properly chosen meshdependent weights, it is shown that the approximations to the solutions of the Stokes equations are of optimal order. The results of some computational experiments are also provided; these illustrate, among other things, the necessity of introducing the weights.
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 S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 3592. MR 0162050 (28:5252)
 [2]
 A. Aziz, R. Kellogg, and A. Stephens, Leastsquares methods for elliptic systems, Math. Comp. 44 (1985), 5370. MR 771030 (86i:65069)
 [3]
 M. A. Behr, L. P. Franca, and T. E. Tezduyar, Stabilized finite element methods for the velocitypressurestress formulation of incompressible flows, Comput. Methods Appl. Mech. Engrg. 104 (1993), 3148. MR 1210650 (94d:76051)
 [4]
 P. Bochev, Leastsquares methods for NavierStokes equations, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1994.
 [5]
 P. Bochev and M. Gunzburger, Accuracy of leastsquares methods for the NavierStokes equations, Comput. & Fluids 22 (1993), 549563. MR 1230751 (94e:76053)
 [6]
 J. H. Bramble and A. H. Schatz, Leastsquares methods for 2mth order elliptic boundary value problems, Math. Comp. 25 (1971), 132. MR 0295591 (45:4657)
 [7]
 F. Brezzi, J. Rappaz, and P.A. Raviart, Finitedimensional approximation of nonlinear problems, Part I: Branches of nonsingular solutions, Numer. Math. 36 (1980), 125. MR 595803 (83f:65089a)
 [8]
 C.L. Chang, A mixed finite element method for the Stokes problem: an accelerationpressure formulation, Appl. Math. Comput. 36 (1990), 135146. MR 1049399 (91d:65168)
 [9]
 , Leastsquares finiteelement method for incompressible flow in 3D (to appear).
 [10]
 C.L. Chang and M. Gunzburger, A finite element method for first order systems in three dimensions, Appl. Math. Comput. 23 (1987), 171184. MR 896976 (89m:65100)
 [11]
 C.L. Chang and B.N. Jiang, An error analysis of leastsquares finite element methods of velocityvorticitypressure formulation for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 84 (1990), 247255. MR 1082823 (91k:76106)
 [12]
 C.L. Chang and L. Povinelli, Piecewise linear approach to the Stokes equations in 3D (to appear). MR 1342362 (97b:76040)
 [13]
 P. Ciarlet, Finite element method for elliptic problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [14]
 G. Fix, M. Gunzburger, R. Nicolaides, and J. Peterson, Mixed finite element approximations for the biharmonic equations, Proc. 5th Internat. Sympos. on Finite Elements and Flow Problems (J. T. Oden, ed.), University of Texas, Austin, 1984, pp. 281286.
 [15]
 L. P. Franca and R. Stenberg, Error analysis of some Galerkin leastsquares methods for the elasticity equations, SIAM J. Numer. Anal. 28 (1991), 16801697. MR 1135761 (92k:73066)
 [16]
 V. Girault and P.A. Raviart, Finite element methods for NavierStokes equations, Springer, Berlin, 1986. MR 851383 (88b:65129)
 [17]
 M. Gunzburger, Finite element methods for viscous incompressible flows, Academic Press, Boston, 1989. MR 1017032 (91d:76053)
 [18]
 M. Gunzburger, M. Mundt, and J. Peterson, Experiences with computational methods for the velocityvorticity formulation of incompressible viscous flows, Computational Methods in Viscous Aerodynamics (T. K. S. Murthy and C. A. Brebbia, eds.), Elsevier, Amsterdam, 1990, pp. 231271. MR 1188660 (93f:76067)
 [19]
 B.N. Jiang, A leastsquares finite element method for incompressible NavierStokes problems, Internat. J. Numer. Methods Fluids 14 (1992), 943859.
 [20]
 B.N. Jiang and C. Chang, Leastsquares finite elements for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 78 (1990), 297311. MR 1039687 (91h:76058)
 [21]
 B.N. Jiang, T. Lin, and L. Povinelli, Largescale computation of incompressible viscous flow by leastsquares finite element methods, Comput. Methods Appl. Mech. Engrg. (to appear). MR 1277480 (95a:76060)
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 B.N. Jiang and L. Povinelli, Leastsquares finite element method for fluid dynamics, Comput. Methods Appl. Mech. Engrg. 81 (1990), 1337. MR 1071091 (91f:76040)
 [23]
 B.N. Jiang and V. Sonnad, Leastsquares solution of incompressible NavierStokes equations with the pversion of finite elements, NASA TM 105203 (ICOMP Report 9114), NASA, Cleveland, 1991.
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 D. Lefebvre, J. Peraire, and K. Morgan, Leastsquares finite element solution of compressible and incompressible flows, Internat. J. Numer. Methods Heat Fluid Flow 2 (1992), 99113.
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 J.L. Lions and E. Magenes, Nonhomogeneous elliptic boundary value problems and applications, Vol. I, Springer, Berlin, 1972.
 [26]
 M. Renardy and R. Rogers, Introduction to partial differential equations, Springer, Berlin, 1993. MR 1211418 (94c:35001)
 [27]
 R. Roitberg and Z. Seftel, A theorem on homeomorphisms for elliptic systems and its applications, Math. USSR Sb. 7 (1969), 439465.
 [28]
 L. Tang and T. Tsang, A leastsquares finite element method for timedependent incompressible flows with thermal convection, Internat. J. Numer. Methods Fluids (to appear).
 [29]
 W. Wendland, Elliptic systems in the plane, Pitman, London, 1979. MR 518816 (80h:35053)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199412575734
PII:
S 00255718(1994)12575734
Article copyright:
© Copyright 1994
American Mathematical Society
