Analysis of least squares finite element methods for the Stokes equations

Authors:
Pavel B. Bochev and Max D. Gunzburger

Journal:
Math. Comp. **63** (1994), 479-506

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 first-order velocity-vorticity-pressure 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 velocity-vorticity-pressure 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 velocity-vorticity-pressure formulation. The second class uses weighted -norms of the residuals to circumvent this flaw. For properly chosen mesh-dependent 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.

**[1]**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), 35–92. MR**0162050****[2]**A. K. Aziz, R. B. Kellogg, and A. B. Stephens,*Least squares methods for elliptic systems*, Math. Comp.**44**(1985), no. 169, 53–70. MR**771030**, 10.1090/S0025-5718-1985-0771030-5**[3]**Marek A. Behr, Leopoldo P. Franca, and Tayfun E. Tezduyar,*Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows*, Comput. Methods Appl. Mech. Engrg.**104**(1993), no. 1, 31–48. MR**1210650**, 10.1016/0045-7825(93)90205-C**[4]**P. Bochev,*Least-squares methods for Navier-Stokes equations*, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1994.**[5]**Pavel B. Bochev and Max D. Gunzburger,*Accuracy of least-squares methods for the Navier-Stokes equations*, Comput. & Fluids**22**(1993), no. 4-5, 549–563. MR**1230751**, 10.1016/0045-7930(93)90025-5**[6]**J. H. Bramble and A. H. Schatz,*Least squares methods for 2𝑚th order elliptic boundary-value problems*, Math. Comp.**25**(1971), 1–32. MR**0295591**, 10.1090/S0025-5718-1971-0295591-8**[7]**F. Brezzi, J. Rappaz, and P.-A. Raviart,*Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions*, Numer. Math.**36**(1980/81), no. 1, 1–25. MR**595803**, 10.1007/BF01395985**[8]**Ching Lung Chang,*A mixed finite element method for the Stokes problem: an acceleration-pressure formulation*, Appl. Math. Comput.**36**(1990), no. 2, 135–146. MR**1049399**, 10.1016/0096-3003(90)90016-V**[9]**-,*Least-squares finite-element method for incompressible flow in*3-*D*(to appear).**[10]**Ching Lung Chang and Max D. Gunzburger,*A finite element method for first order elliptic systems in three dimensions*, Appl. Math. Comput.**23**(1987), no. 2, 171–184. MR**896976**, 10.1016/0096-3003(87)90037-3**[11]**Ching Lung Chang and Bo-Nan 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), no. 3, 247–255. MR**1082823**, 10.1016/0045-7825(90)90079-2**[12]**Ching Lung Chang,*Piecewise linear approach to the Stokes equations in 3-D*, Appl. Math. Comput.**72**(1995), no. 1, 61–75. MR**1342362**, 10.1016/0096-3003(94)00176-5**[13]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[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. 281-286.**[15]**Leopoldo P. Franca and Rolf Stenberg,*Error analysis of Galerkin least squares methods for the elasticity equations*, SIAM J. Numer. Anal.**28**(1991), no. 6, 1680–1697. MR**1135761**, 10.1137/0728084**[16]**Vivette Girault and Pierre-Arnaud Raviart,*Finite element methods for Navier-Stokes equations*, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR**851383****[17]**Max D. Gunzburger,*Finite element methods for viscous incompressible flows*, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. A guide to theory, practice, and algorithms. MR**1017032****[18]**M. Gunzburger, M. Mundt, and J. Peterson,*Experiences with finite element methods for the velocity-vorticity formulation of three-dimensional, viscous, incompressible flows*, Computational methods in viscous aerodynamics, Elsevier, Amsterdam, 1990, pp. 231–271. MR**1188660****[19]**B.-N. Jiang,*A least-squares finite element method for incompressible Navier-Stokes problems*, Internat. J. Numer. Methods Fluids**14**(1992), 943-859.**[20]**Bo-Nan Jiang and C. L. Chang,*Least-squares finite elements for the Stokes problem*, Comput. Methods Appl. Mech. Engrg.**78**(1990), no. 3, 297–311. MR**1039687**, 10.1016/0045-7825(90)90003-5**[21]**Bo-Nan Jiang, T. L. Lin, and Louis A. Povinelli,*Large-scale computation of incompressible viscous flow by least-squares finite element method*, Comput. Methods Appl. Mech. Engrg.**114**(1994), no. 3-4, 213–231. MR**1277480**, 10.1016/0045-7825(94)90172-4**[22]**Bo-Nan Jiang and Louis A. Povinelli,*Least-squares finite element method for fluid dynamics*, Comput. Methods Appl. Mech. Engrg.**81**(1990), no. 1, 13–37. MR**1071091**, 10.1016/0045-7825(90)90139-D**[23]**B.-N. Jiang and V. Sonnad,*Least-squares solution of incompressible Navier-Stokes equations with the p-version of finite elements*, NASA TM 105203 (ICOMP Report 91-14), NASA, Cleveland, 1991.**[24]**D. Lefebvre, J. Peraire, and K. Morgan,*Least-squares finite element solution of compressible and incompressible flows*, Internat. J. Numer. Methods Heat Fluid Flow**2**(1992), 99-113.**[25]**J.-L. Lions and E. Magenes,*Nonhomogeneous elliptic boundary value problems and applications*, Vol. I, Springer, Berlin, 1972.**[26]**Michael Renardy and Robert C. Rogers,*An introduction to partial differential equations*, Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 1993. MR**1211418****[27]**R. Roitberg and Z. Seftel,*A theorem on homeomorphisms for elliptic systems and its applications*, Math. USSR Sb.**7**(1969), 439-465.**[28]**L. Tang and T. Tsang,*A least-squares finite element method for time-dependent incompressible flows with thermal convection*, Internat. J. Numer. Methods Fluids (to appear).**[29]**W. L. Wendland,*Elliptic systems in the plane*, Monographs and Studies in Mathematics, vol. 3, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR**518816**

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1994-1257573-4

Article copyright:
© Copyright 1994
American Mathematical Society