# zbMATH — the first resource for mathematics

A least squares Galerkin-Petrov nonconforming mixed finite element method for the stationary conduction-convection problem. (English) Zbl 1189.76345
The authors consider a steady-state conduction-convection problem in two space dimensions, which corresponds to incompressible Navier-Stokes equations in Boussinesq approximation. The set of partial differential equations is discretized employing a least square formulation together with a Petrov-Galerkin method which uses non-conforming finite elements on triangular meshes. Different polynomial degrees are used for velocity, temperature and pressure. For the latter linear elements are applied, while for the two former variables quadratic ones are utilized. Existence and uniqueness of the discrete solution are proven together with error estimates which are optimal, supposing sufficiently large viscosity. Furthermore, it is shown that the mixed finite element spaces do not need to satisfy the inf-sup condition.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76R10 Free convection 80A20 Heat and mass transfer, heat flow (MSC2010) 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text:
##### References:
  Carey, G.; Oden, J., Finite elements: fluid mechanics, VIV, (1986), Prentice-Hall  Cuvelier, C.; Segal, A.; Steenhoven, A., Finite element methods and navier – stokes equations, (1986), D Reidei Publishing Company · Zbl 0649.65059  DiBenedetto, E.; Friedman, A., Conduction – convection problems problems with change of phase, J. differential equation, 62, 129-185, (1986) · Zbl 0593.35085  Zeider, E., Nonlinear functional analysis and applications IV, ()  France, L.; Hughes, T., Two classes of mixed finite element methods, Comput. meth. appl. mech. engrg., 69, 225-235, (1989)  Hughes, T.; Tezduyar, T.; Balestra, M., A new finite element formulation for computational fluid dynamics. V. circumventing the babuska – brezzi condition: astable petrov – galerkin formulation of the Stokes problem accommodating equal-order interpolation, Comput. meth. appl. mech. engrg., 59, 85-99, (1986) · Zbl 0622.76077  Hughes, T.; Tezduyar, T., A new finite element formulation for computational fluid dynamics. VII. the Stokes problem with various well posed boundary conditions, symmetric formulations that converge for all velocity/pressure spaces, Comput. meth. appl. mech. engrg., 65, 85-96, (1987) · Zbl 0635.76067  Brezzi, F.; Douglas, J., Stabilized mixed methods for the Stokes problem, Numer. math., 53, 225-235, (1988) · Zbl 0669.76052  Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. comp., 52, 495-508, (1989) · Zbl 0669.76051  Hughes, T.; Tezduyar, T., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. meth. appl. mech. engrg., 45, 217-284, (1984) · Zbl 0542.76093  Johnson, C.; Saranen, J., Streamline diffusion methods for the incompressible Euler and navier – stokes equations, Math. comp., 47, 1-18, (1986) · Zbl 0609.76020  Fu, D., The hydromechanics value simulates, (1994), National Defence Industry Press Beijing, (in Chinese)  Liu, R.; Shu, Q., Some new methods of the computation hydromechanics, (2003), Science Press Beijing, (in Chinese)  Xin, X.; Liu, R.; Jiang, B., The computation hydrodynamics, (1994), National University of Defense Technology Press Changsha, (in Chinese)  Zhou, T.; Feng, M., A least squares galerkin – petrov finite element method for the stationary navier – stokes equations, Math. comp., 60, 202, 531-543, (1993) · Zbl 0778.65081  Luo, Z., Theory bases and applications of finite element mixed methods, (2006), Science Press Beijing, (in Chinese)  Adams, R., Sobolev spaces, (1975), Academic Press New York · Zbl 0314.46030  Ammi, A.; Marion, M., Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the navier – stokes equations, Numer. math., 68, 2, 189-213, (1994) · Zbl 0811.76035  Li, K.; Zhou, L., Finite element nonlinear Galerkin methods for penalty navier – stokes equations, Math. numer. sin., 17, 4, 360-380, (1995), (in Chinese) · Zbl 0859.76040  Lou, Z.; Wang, L., Nonlinear Galerkin mixed element methods for the non stationary conduction – convection problems(I): the continuous-time case, Math. numer. sin., 20, 3, 283-304, (1998), (in Chinese)  Temam, R., Navier – stokes equation, theory and numerical analysis, (1984), North-Hoolland New York, Amstedam  Girault, V.; Raviart, P., Finite element method for navier – stokes equations: theory and algorithms, (1986), Springer-Verlag New York  Shen, S., The finite element analysis for the conduction – convection problems, Math. numer. sin., 2, 170-182, (1994), (in Chinese) · Zbl 0922.76105  Shi, D.; Ren, J., Nonconforming mixed finite element approximation to the stationary navier – stokes equations on anisotropic meshes, Nonlinear anal., 71, 9, 3842-3852, (2009) · Zbl 1166.76030  Shi, D.; Ren, J., Nonconforming mixed finite element method for the stationary conduction – convection problem, Inter. J. numer. anal. model., 6, 2, 293-310, (2009) · Zbl 1165.65080  Zhu, Q.; Lin, Q., Superconvergence of finite element theory, (1989), Hunan Science and Technology Press Changsha, (in Chinese)  Lee, H.; Sheen, D., A new quadratic nonconforming finite element on rectangles, Numer. meth. PDEs., 22, 4, 954-970, (2005) · Zbl 1097.74059  Cai, W., Convergence of two nonconforming memerane finite elements, Math. numer. sin., 1, 63-74, (1986), (in Chinese) · Zbl 0636.73064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.