Geodesic
On the stability analysis of flows in channels of elliptic cross section using the finite element method on unstructured meshes
Numerical methods and programming, Tome 21 (2020) no. 4, pp. 373-387.

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The existing technique for the numerical analysis of incompressible fluid flow stability in channels of constant cross section was earlier extended to the case of local spatial approximations on unstructured meshes, which leads to large sparse matrices. The finite element method is employed for spatial approximation and a new efficient Newton-type method is used to solve partial eigenvalues problems arising in flow stability analysis. A detailed numerical study of the proposed approach is carried out in this paper by the example of Poiseuille flow in a channel of elliptic cross section. Performance ability of the approach is demonstrated for a wide range of the cross-sectional semiaxes ratio, including the case of linear instability of the flow under consideration. The convergence of the leading part of the spectrum with respect to the grid size is shown. Our results are in good agreement with those obtained via approximation by the spectral collocation method.
Keywords: hydrodynamic stability; finite element method; unstructured mesh; sparse matrices; channel of elliptic cross-section; partial eigenvalue problem; inexact Newton method. 
@article{VMP_2020_21_4_a2,
     author = {N. V. Klyushnev},
     title = {On the stability analysis of flows in channels of elliptic cross section using the finite element method on unstructured meshes},
     journal = {Numerical methods and programming},
     pages = {373--387},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2020_21_4_a2/}
}
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N. V. Klyushnev. On the stability analysis of flows in channels of elliptic cross section using the finite element method on unstructured meshes. Numerical methods and programming, Tome 21 (2020) no. 4, pp. 373-387. http://geodesic.mathdoc.fr/item/VMP_2020_21_4_a2/