Numerical simulation of complex\ud viscoelastic flows using discontinuous\ud galerkin spectral/hp element methods

Subject: QA

References
(39)
3.10. One dimensional advection of square wave on domain Ω = [0, 10] subdivided into 10 equallysized elements. . . . . . . . . . . . . . . . . . . . .
3.11. Comparison of Galerkin projection for the Gaussian function of ((a), (b)) spectral/hp element method and ((c), (d)) linear finite element method for DOF = 90 and the corresponding L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12. Comparison of Galerkin projection for the hat function of ((a), (b)) spectral/hp element method and ((c), (d)) linear finite element method for DOF = 90 and the corresponding L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.13. Comparison of Galerkin projection for the square wave function of ((a), (b)) spectral/hp element method and ((c), (d)) linear finite element method for DOF = 90 and the corresponding L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14. Comparison of the numerical results for the advection equation after 30000 timesteps with Δt = 10−4 using the continuous Galerkin method and the discontinuous Galerkin method for the spectral/hp element method for P = 8, Nel = 10 ((a), (b)) and the linear finite element method, i.e. P = 1, for Nel = 45 ((c), (d)) for the smooth Gaussian function and the L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . .
3.15. Comparison of the numerical results for the advection equation after 30000 timesteps with Δt = 10−4 using the continuous Galerkin method and the discontinuous Galerkin method for the spectral/hp element method for P = 8, Nel = 10 ((a), (b)) and the linear finite element method, i.e. P = 1, for Nel = 45 ((c), (d)) for the hat function and the L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . .
3.16. Comparison of the numerical results for the advection equation after 30000 timesteps with Δt = 10−4 using the continuous Galerkin method and the discontinuous Galerkin method for the spectral/hp element method for P = 8, Nel = 10 ((a), (b)) and the linear finite element method, i.e. P = 1, for Nel = 45 ((c), (d)) for the hat function and the L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . .
F. Brezzi. On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers. RAIRO Anal. Numér., 8:129151, 1974.
A.N. Brooks and T.J.R. Hughes. Streamline upwind/PetrovGalerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations. Comput. Meth. Appl. Mech. Eng., 32(1):199259, 1982.
R.A. Brown, M.J. Szady, P.J. Northey, and R.C. Armstrong. On the numerical stability of mixed finiteelement methods for viscoelastic flows governed by differential constitutive equations. Theor. Comput. Fluid Dyn., 5(2):77106, 1993.

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