Abstract
In this paper, we consider a stabilized finite element method for the approximation of the Stokes eigenvalue problem on triangular domains. The method depends on orthogonal subscales that has proved to be an appropriate means for approximating eigenvalue problems in the framework of residual based approaches. We consider several isosceles triangular domains with various apex angles to investigate the characteristics of the eigensolutions in regards to the variation of the domain properties. This study presents the first finite element approximation to the solutions of the Stokes eigenvalue problem on triangular domains, to the best of our knowledge. We provide plots of several velocity and pressure fields corresponding to the fundamental eigenmodes to analyze the flow characteristics in detail. Furthermore, we consider the problem on triangular domains including a crack, and investigate the influence of the length of the slit on the fundamental mode to some extent. The results reveal the correlation between the domain properties and the eigenpairs, and the fact that there are various critical lengths of the slit where the eigenspace is notably affected.
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References
Foias C, Manley O, Rosa R, Temam R (2001) Navier-Stokes equations and turbulence. Encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge
Nerli A, Camarri S (2006) Stokes eigenfunctions and galerkin projection of the disturbance equations in plane poiseuille flow: a systematic analytical approach. Meccanica 41(6):671–680
Schneider K, Farge M (2008) Final states of decaying 2D turbulence in bounded domains: Influence of the geometry. Physica D: Nonlinear Phenomena 237(14–17):2228–2233
Labrosse G, Leriche E, Lallemand P (2014) Stokes eigenmodes in cubic domain: their symmetry properties. Theor Comput Fluid Dyn 28(3):335–356
Leriche E, Labrosse G (2000) High-order direct Stokes solvers with or without temporal splitting: numerical investigations of their comparative properties. SIAM J Sci Comput 22(4):1386–1410
Leriche E, Labrosse G (2011) Are there localized eddies in the trihedral corners of the Stokes eigenmodes in cubical cavity? Comput Fluids 43(1):98–101
Gedicke J, Khan A (2018) Arnold-Winther mixed finite elements for Stokes eigenvalue problems. SIAM J Sci Comput 40(5):A3449–A3469
Han J, Zhang Z, Yang Y (2015) A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem. Numer Methods Partial Differ Equ 31(1):31–53
Türk Ö, Boffi D, Codina R (2016) A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems. Comput Methods Appl Mech Eng 310:886–905
John V, Kaiser K, Novo J (2016) Finite element methods for the incompressible Stokes equations with variable viscosity. ZAMM J Appl Math Mech 96(2):205–216
Du S, Duan H (2018) Analysis of a stabilized finite element method for Stokes equations of velocity boundary condition and of pressure boundary condition. J Comput Appl Math 337:290–318
Pearson JW, Pestana J, Silvester DJ (2018) Refined saddle-point preconditioners for discretized Stokes problems. Numer Math 138(2):331–363
Santo ND, Deparis S, Manzoni A, Quarteroni A (2018) Multi space reduced basis preconditioners for parametrized Stokes equations. Comput Math Appl 77:1583–1604
Cioncolini A, Boffi D (2019) The mini mixed finite element for the Stokes problem: an experimental investigation. Comput Math Appl 77(9):2432–2446
Prato Torres R, Domínguez C, Díaz S (2019) An adaptive finite element method for a time-dependent Stokes problem. Numer Methods Partial Differ Equ 35(1):325–348
Bustamante C, Power H, Florez W (2013) A global meshless collocation particular solution method for solving the two-dimensional Navier-Stokes system of equations. Comput Math Appl 65(12):1939–1955
Li M, Tang T (1996) Steady viscous flow in a triangular cavity by efficient numerical techniques. Comput Math Appl 31(10):55–65
Chen L, Labrosse G, Lallemand P, Luo LS (2016) Spectrally accurate Stokes eigen-modes on isosceles triangles. Comput Fluids 132:1–9
Cao J, Wang Z, Chen L (2017) A triangular spectral method for the Stokes eigenvalue problem by the stream function formulation. Numer Methods Partial Differ Equ 34(3):825–837
Yolcu SY, Yolcu T (2012) Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators. J Math Phys 53(4):043508
Babuška I, Osborn J (1991) Eigenvalue problems. In: Finite element methods (part 1), Handbook of numerical analysis, vol 2. Elsevier, Amsterdam, pp 641–787
Codina R (2000) Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput Methods Appl Mech Eng 190:1579–1599
Codina R (2008) Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales. Appl Numer Math 58:264–283
Codina R, Blasco J (1997) A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput Methods Appl Mech Eng 143:373–391
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Türk, Ö. Approximation of the Stokes eigenvalue problem on triangular domains using a stabilized finite element method . Meccanica 55, 2021–2031 (2020). https://doi.org/10.1007/s11012-020-01243-w
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DOI: https://doi.org/10.1007/s11012-020-01243-w