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Computation of cusp singularities for operator equations and their discretizations. (English) Zbl 0672.65031
Authors’ summary: We discuss the direct calculation of cusp singularities as solutions of a minimally augmented defining system, which is nonsingular under the canonical cusp conditions. The underlying operator equation may have any finite Fredholm index, and bounds on the discretization error are derived for the case of projection methods.
Reviewer: J.Kolomý

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J25 Iterative procedures involving nonlinear operators
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[1] Beyn, W.J., Defining equations for singular solutions and numerical applications, (), 42-56
[2] Fink, J.P.; Rheinboldt, W.C., A geometric framework for the numerical study of singular points, SIAM J. numer. anal., 24, 618-633, (1987) · Zbl 0623.65065
[3] Golubitsky, M.; Keyfitz, B.L., A qualitative study of the steady-state solutions for a continuous flow stirred tank chemical reactor, SIAM J. math. anal., 11, 316-361, (1980) · Zbl 0448.58001
[4] Golubitsky, M.; Schaeffer, D.G., Singularities and groups in bifurcation theory, (1985), Springer New York · Zbl 0607.35004
[5] Griewank, A.; Rabier, P., Critical points of mixed fluids and their numerical treatment, (), 90-97 · Zbl 0635.76099
[6] Griewank, A.; Reddien, G.W., Characterization and computation of generalized turning points, SIAM J. numer. anal., 21, 176-185, (1984) · Zbl 0536.65031
[7] Griewank, A.; Reddien, G.W., The approximation of generalized turning points by projection methods with superconvergence to the critical parameter, Numer. math., 48, 591-606, (1986) · Zbl 0596.65040
[8] Jepson, A.D.; Spence, A., Folds in solutions of two-parameter systems and their calculation, SIAM J. numer. anal., 22, 347-368, (1985), Part 1 · Zbl 0576.65052
[9] Jepson, A.D.; Spence, A., Singular points and their computation, (), 195-209 · Zbl 0579.65048
[10] Spence, A.; Jepson, A.D., The numerical calculation of cusps, bifurcation points and isola formation points in two-parameter problems, (), 502-514 · Zbl 0547.65047
[11] A.D. Jepson and A. Spence, The numerical solution of nonlinear equations having several parameters II: vector equations, SIAM J. Numer. Anal., to appear. · Zbl 0597.65051
[12] Kunkel, P., Efficient computation of singular points, () · Zbl 0676.65048
[13] Moore, G.; Spence, A., The calculation of turning points of nonlinear equations, SIAM J. numer. anal., 17, 567-576, (1980) · Zbl 0454.65042
[14] Peng, D.Y.; Robinson, D.B., A new two-constant equation of state, Ind. eng. chem. fundamentals, 59-64, (1976)
[15] Pönisch, G., Computing hysteresis points of nonlinear equations depending on two parameters, Computing, 39, 1-17, (1987) · Zbl 0613.65053
[16] Rheinboldt, W.C., Computation of critical boundaries on equilibrium manifolds, SIAM J. numer. anal., 19, 653-669, (1982) · Zbl 0489.65033
[17] Roose, D.; Caulewarts, R., Direct methods for the computation of a nonsimple turning point corresponding to a cusp, (), 426-440
[18] Roose, D.; de Dier, B., Numerical determination of an emanating branch of Hopf bifurcation points in a two parameter problem, () · Zbl 0677.65090
[19] Roose, D.; Piessens, R., Numerical computation of turning points and cusps, Numer. math., 46, 189-211, (1985) · Zbl 0543.65040
[20] Seydel, R., Numerical computation of branch points in nonlinear equations, Numer. math., 33, 339-352, (1979) · Zbl 0396.65023
[21] Spence, A.; Werner, B., Non-simple turning points and cusps, IMA J. numer. anal., 2, 413-427, (1982) · Zbl 0539.65043
[22] Yang, Zhong-Hua; Keller, H.B., A direct method for computing higher order folds, SIAM J. sci. stat. comput., 7, 351-361, (1986) · Zbl 0618.65049
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