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Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations. (English) Zbl 1067.76562

Summary: We formulate and study finite element methods for the solution of the incompressible Stokes equations based on the application of a least-squares minimization principle to an equivalent first order velocity-pressure-stress system. Our least-squares functional involves the \(L^ 2\)-norms of the residuals of each equation multiplied by a mesh-dependent weight. Each weight is determined according to the Agmon-Douglis-Nirenberg index of the corresponding equation. As a result, the approximating spaces are not subject to the LBB condition and conforming discretizations are possible with merely continuous finite element spaces. Moreover, the resulting discrete problems involve only symmetric, positive definite systems of linear equations, i.e. assembly of the discretization matrix is not required even at the element level. We prove that the least-squares approximations converge to the solutions of the Stokes problem at the best possible rate and then present some numerical examples illustrating our theoretical results. Among other things, these numerical examples indicate that the method is not optimal without the weights in the least-squares functional.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
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[1] Bochev, P., Least-squares methods for the Stokes and Navier-Stokes equations, () · Zbl 0970.35098
[2] Bochev, P.; Gunzburger, M., Analysis of least-squares finite element methods for the Stokes equations, Math. comput., 63, 479-506, (1994) · Zbl 0816.65082
[3] Bochev, P.; Gunzburger, M., Analysis of weighted least-squares finite element method for the Navier-Stokes equations, (), 584-587
[4] Bochev, P.; Gunzburger, M., Accuracy of least-squares methods for the Navier-Stokes equations, Comput. fluids, 22, 549-563, (1993) · Zbl 0779.76039
[5] Chang, C.-L., A mixed finite element method for the Stokes problem: an acceleration-pressure formulation, Appl. math. comput., 36, 135-146, (1990) · Zbl 0702.76075
[6] C.-L. Chang, Least-squares finite-element method for incompressible flow in 3-D, in preparation.
[7] Chang, C.-L.; Jiang, B.-N., An error analysis of least-squares finite element methods of velocity-vorticity-pressure formulation for the Stokes problem, Comput. methods appl. mech. engrg., 84, 247-255, (1990) · Zbl 0733.76042
[8] C.-L. Chang and L. Povinelli, Piecewise linear approach to the Stokes equations in 3-D, in preparation.
[9] Jiang, B.-N., A least-squares finite element method for incompressible Navier-Stokes problems, Int. J. numer. methods fluids, 14, 843-859, (1992) · Zbl 0753.76097
[10] Jiang, B.-N.; Chang, C., Least-squares finite elements for the Stokes problem, Comput. methods appl. mech. engrg., 78, 297-311, (1990) · Zbl 0706.76033
[11] B.-N. Jiang, T. Lin and L. Povinelli, Large-scale computation of incompressible viscous flow by least-squares finite element method, Comput. Methods Appl. Mech. Engrg., in preparation.
[12] Jiang, B.-N.; Povinelli, L., Least-squares finite element method for fluid dynamics, Comput. methods appl. mech. engrg., 81, 13-37, (1990) · Zbl 0714.76058
[13] Lefebvre, D.; Peraire, J.; Morgan, K., Least-squares finite element solution of compressible and incompressible flows, Int. J. num. methods heat fluid flow, 2, 99-113, (1992)
[14] Girault, V.; Raviart, P.-A., Finite element methods for Navier-Stokes equations, (1986), Springer Berlin · Zbl 0413.65081
[15] Gunzburger, M., Finite element methods for viscous incompressible flows, (1989), Academic Press Boston, MA · Zbl 0697.76031
[16] Aziz, A.; Kellogg, R.; Stephens, A., Least-squares methods for elliptic systems, Math. comput., 44, 53-70, (1985) · Zbl 0609.35034
[17] Johnson, C., A mixed finite element method for the Navier-Stokes equations, Rairo m^{2}an, 12, 335-348, (1978) · Zbl 0399.76035
[18] Behr, M.A.; Franca, L.P.; Tezduyar, I.E., Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows, Comput. methods appl. mech. engrg., 104, 31-48, (1993) · Zbl 0771.76033
[19] Franca, L.P.; Stenberg, R., Error analysis of some Galerkin least-squares methods for the elasticity equations, SIAM J. numer. anal., 28, 1680-1697, (1991) · Zbl 0759.73055
[20] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Commun. pure appl. math., 17, 35-92, (1964) · Zbl 0123.28706
[21] Bramble, J.H.; Schatz, A.H., Least-squares methods for 2mth order elliptic boundary value problems, Math. comput., 25, 1-32, (1971) · Zbl 0216.49202
[22] L. Tang and T. Tsang, A least-squares finite element method for time-dependent incompressible flows with thermal convection, Int. J. Numer. Methods Fluids, in preparation. · Zbl 0779.76045
[23] Wendland, W., Elliptic systems in the plane, (1979), Pitman London · Zbl 0396.35001
[24] Gunzburger, M.; Mundt, M.; Peterson, J., Experiences with computational methods for the velocity-vorticity formulation of incompressible viscous flows, (), 231-271
[25] Roitberg, J.; Seftel, Z., A theorem on homeomorphisms for elliptic systems and its applications, Math. USSR sb., 7, 439-465, (1969)
[26] Lions, J.-L.; Magenes, E., ()
[27] Ciarlet, P., Finite element method for elliptic problems, (1978), North-Holland Amsterdam
[28] Chang, C.-L.; Gunzburger, M., A finite element method for first order systems in three dimensions, Appl. math. comput., 23, 171-184, (1987) · Zbl 0631.65108
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