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block matrices; eigenvalues; Cayley transform; Navier-Stokes; large sparse generalized eigenvalue problems; Hopf bifurcations
This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalized eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem. Numerical experiments are performed on large matrices arising from a discretization of the flow over a backward facing step.
[1] W. E. Arnoldi: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9 (1951), 17-29. DOI 10.1090/qam/42792 | MR 0042792 | Zbl 0042.12801
[2] K. N. Christodoulou, L.E. Scriven: Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem. J. Sci. Comput. 3 (1988), 355-406. DOI 10.1007/BF01065178 | Zbl 0677.65032
[3] K. A. Cliffe, T .J. Garratt, and A. Spence: Iterative methods for the detection of hopf bifurcations in finite element discretizations of incompressible flow problems. Submitted SIAM J. Sci. Stat. Соmр. (1992).
[4] K .A. Cliffe T. J. Garratt A. Spence, and K. H. Winters: Eigenvalue calculations for flow over a backward facing step. AEA Decommissioning and Rad waste report, 1992.
[5] J. N. Franklin: Matrix theory. Prentice-Hall, New Jersey, 1968. MR 0237517 | Zbl 0174.31501
[6] T. J. Garratt G. Moore, and A. Spence: Two methods for the numerical detection of Hopf bifurcations. In: Bifurcation and chaos: analysis, algorithms, applications (R. Seydel, F.W. Schneider, and H. Troger, eds.), Birkhäuser, 1991, pp. 119-123. MR 1109515
[7] D. K. Gartling: A test problem for outflow boundary conditions - flow over a backward facing step. Int. J. Num. Methods Fluids 11 (1990), 953-967. DOI 10.1002/fld.1650110704
[8] G. H. Golub, C. F. van Loan: Matrix Computations. The Johns Hopkins University Press, Baltimore, Maryland, 1983. MR 0733103 | Zbl 0559.65011
[9] C. P. Jackson: A finite element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182 (1987), 23-45. DOI 10.1017/S0022112087002234 | Zbl 0639.76041
[10] D. S. Malkus: Eigenproblems associated with the discrete LBB condition for incompressible finite elements. Int. J. for Eng. Sci. 19(1981), 1299-1310. MR 0660563 | Zbl 0457.73051
[11] G. Peters, J. H. Wilkinson: Inverse iteration, illconditioned equations and Newton's method. SIAM Rev. 21 (1979), 339-360. DOI 10.1137/1021052 | MR 0535118
[12] G. W. Stewart: Simultaneous iteration for computing invariant subspaces of nonhermitian matrices. Numer. Math. 25 (1976), 123-136. DOI 10.1007/BF01462265 | MR 0400677
[13] J. H. Wilkinson: The algebraic eigenvalue problem. Claredon Press, Oxford, 1965. MR 0184422 | Zbl 0258.65037
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