This research monograph concerns the design of efficient, robust and reliable algorithms for the numerical simulation of multiscale phenomena with orientation for applications. Various modern techniques from scattered data modeling, such as splines over triangulation and radial basis functions, are combined with customized adaptive strategies, which are developed individually in this work.
Very recent results concerning the approximation order and the numerical stability of polyharmonic spline interpolation [see J. Duchon, R.A.I.R.O. Anal. Numér. 12, 325–334 (1978; Zbl 0403.41003)] are renewed. Recent developments concerning thinning algorithms with emphasis on their application to terrain modeling and image compression are given various alternative multilevel approximation methods for bivariate scattered date are also presented [see M. S. Floater and A. Iske, J. Comput. Appl. Math 73, No.1–2, 65–78 (1996; Zbl 0859.65006)]. Extensive numerical examples, mainly arising from real world application are given.
Meshfree recent methods, which are discretization schemes for partial differential equation are also discussed. Special emphasis is given to a comparison between the various numerical algorithms developed here and comparable state-of-the-art methods.