Special Numerical Analysis/Scientific Computing Seminar
Abstract: Nash iteration was introduced by J. Nash [Ann. Math. 63 (1956), 20--63] as a technique to embed Riemannian manifolds in Euclidean space. It was later developed by Moser and Schwartz, and formulated as a generalized implicit function theorem (see L. Nirenberg, Topics in Nonlinear Functional Analysis, NYU, 1973). Central to this setting is a loss of derivatives'. An interpretation in terms of the use of numerical methods was given by the speaker [Numer. Math. 47 (1985), 123--138]. In this talk, recent results will be reviewed, including numerical experiments based upon collocation and radial basis functions.
In the second part of the talk, a trapping principle will be presented for piecewise linear finite element solutions to semilinear elliptic systems, with mixed boundary conditions. The trapping region is one for which the system vector field is outward pointing. Some examples will be presented.
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