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Bubble-enriched least-squares finite element method for transient advective transport. (English) Zbl 1163.65068

Summary: The least-squares finite element method (LSFEM) has received increasing attention in recent years due to advantages over the Galerkin finite element method. The method leads to a minimization problem in the \(L_{2}\)-norm and thus results in a symmetric and positive definite matrix, even for first-order differential equations. In addition, the method contains an implicit streamline upwinding mechanism that prevents the appearance of oscillations that are characteristic of the Galerkin method. Thus, the least-squares approach does not require explicit stabilization and the associated stabilization parameters required by the Galerkin method.
A new approach, the bubble enriched least-squares finite element method (BELSFEM), is presented and compared with the classical LSFEM. The BELSFEM requires a space-time element formulation and employs bubble functions in space and time to increase the accuracy of the finite element solution without degrading computational performance. We apply the BELSFEM and classical least-squares finite element methods to benchmark problems for 1D and 2D linear transport. The accuracy and performance are compared.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
35L45 Initial value problems for first-order hyperbolic systems
65Y20 Complexity and performance of numerical algorithms
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:

[1] B.-N. Jiang, The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics, Scientific Computation, Springer, Berlin, Germany, 1998. · Zbl 0904.76003
[2] I. Christie, D. F. Griffiths, A. R. Mitchell, and O. C. Zienkiewicz, “Finite element methods for second order differential equations with significant first derivatives,” International Journal for Numerical Methods in Engineering, vol. 10, no. 6, pp. 1389-1396, 1976. · Zbl 0342.65065 · doi:10.1002/nme.1620100617
[3] J. J. Westerink and D. Shea, “Consistent higher degree Petrov-Galerkin methods for the solution of the transient convection-diffusion equation,” International Journal for Numerical Methods in Engineering, vol. 28, no. 5, pp. 1077-1101, 1989. · Zbl 0679.76097 · doi:10.1002/nme.1620280507
[4] A. N. Brooks and T. J. R. Hughes, “Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 32, no. 1-3, pp. 199-259, 1982. · Zbl 0497.76041 · doi:10.1016/0045-7825(82)90071-8
[5] J. Donea, “A Taylor-Galerkin method for convective transport problems,” International Journal for Numerical Methods in Engineering, vol. 20, no. 1, pp. 101-119, 1984. · Zbl 0524.65071 · doi:10.1002/nme.1620200108
[6] B.-N. Jiang and L. A. Povinelli, “Least-squares finite element method for fluid dynamics,” Computer Methods in Applied Mechanics and Engineering, vol. 81, no. 1, pp. 13-37, 1990. · Zbl 0714.76058 · doi:10.1016/0045-7825(90)90139-D
[7] B.-N. Jiang, T. L. Lin, and L. A. Povinelli, “Large-scale computation of incompressible viscous flow by least-squares finite element method,” Computer Methods in Applied Mechanics and Engineering, vol. 114, no. 3-4, pp. 213-231, 1994. · doi:10.1016/0045-7825(94)90172-4
[8] J. Donea and L. Quartapelle, “An introduction to finite element methods for transient advection problems,” Computer Methods in Applied Mechanics and Engineering, vol. 95, no. 2, pp. 169-203, 1992. · Zbl 0772.76035 · doi:10.1016/0045-7825(92)90139-B
[9] G. F. Carey and B.-N. Jiang, “Least-squares finite elements for first-order hyperbolic systems,” International Journal for Numerical Methods in Engineering, vol. 26, no. 1, pp. 81-93, 1988. · Zbl 0641.65080 · doi:10.1002/nme.1620260106
[10] C. W. Li, “Least-squares characteristics and finite elements for advection-dispersion simulation,” International Journal for Numerical Methods in Engineering, vol. 29, no. 6, pp. 1343-1358, 1990. · Zbl 0728.73055 · doi:10.1002/nme.1620290615
[11] N.-S. Park and J. A. Liggett, “Taylor-least-squares finite element for two-dimensional advection-dominated unsteady advection-diffusion problems,” International Journal for Numerical Methods in Fluids, vol. 11, no. 1, pp. 21-38, 1990. · Zbl 0696.76105 · doi:10.1002/fld.1650110103
[12] N.-S. Park and J. A. Liggett, “Application of Taylor-least squares finite element to three-dimensional advection-diffusion equation,” International Journal for Numerical Methods in Fluids, vol. 13, no. 6, pp. 759-773, 1991. · Zbl 0739.76033 · doi:10.1002/fld.1650130607
[13] H. Nguyen and J. Reynen, “A space-time least-square finite element scheme for advection-diffusion equations,” Computer Methods in Applied Mechanics and Engineering, vol. 42, no. 3, pp. 331-342, 1984. · Zbl 0517.76089 · doi:10.1016/0045-7825(84)90012-4
[14] T. J. R. Hughes, L. P. Franca, and G. M. Hulbert, “A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations,” Computer Methods in Applied Mechanics and Engineering, vol. 73, no. 2, pp. 173-189, 1989. · Zbl 0697.76100 · doi:10.1016/0045-7825(89)90111-4
[15] K. S. Surana and J. S. Sandhu, “Investigation of diffusion in p-version ‘LSFE’ and ‘STLSFE’ formulations,” Computational Mechanics, vol. 16, no. 3, pp. 151-169, 1995. · Zbl 0835.76050 · doi:10.1007/BF00369778
[16] C. Baiocchi, F. Brezzi, and L. P. Franca, “Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.),” Computer Methods in Applied Mechanics and Engineering, vol. 105, no. 1, pp. 125-141, 1993. · Zbl 0772.76033 · doi:10.1016/0045-7825(93)90119-I
[17] F. Brezzi, L. P. Franca, and A. Russo, “Further considerations on residual-free bubbles for advective-diffusive equations,” Computer Methods in Applied Mechanics and Engineering, vol. 166, no. 1-2, pp. 25-33, 1998. · Zbl 0934.65126 · doi:10.1016/S0045-7825(98)00080-2
[18] L. P. Franca, A. Nesliturk, and M. Stynes, “On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method,” Computer Methods in Applied Mechanics and Engineering, vol. 166, no. 1-2, pp. 35-49, 1998. · Zbl 0934.65127 · doi:10.1016/S0045-7825(98)00081-4
[19] G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for CFD, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, NY, USA, 1999. · Zbl 0954.76001
[20] J. Donea and A. Huerta, Finite Element Methods for Flow Problems, John Wiley & Sons, Chichester, UK, 2003.
[21] C.-C. Yu and J. C. Heinrich, “Petrov-Galerkin method for multidimensional, time-dependent, convective-diffusion equations,” International Journal for Numerical Methods in Engineering, vol. 24, no. 11, pp. 2201-2215, 1987. · Zbl 0636.65117 · doi:10.1002/nme.1620241112
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