Seminario CUICBAS: Global Spectral Methods in Complicated Geometry

Seminario CUICBAS. Por John P. Boyd, University of Michigan. Viernes, 22 de octubre, 2010. Auditorio de la Facultad de Ciencias.

Abstract: Global spectral methods have been successful in many branches of science and engineering. The global weather forecasting model of the United States, for example, is a spherical harmonic spectral model in latitude and longitude. However, single domain spectral methods have been almost exclusively restricted to domains which are a tensor product (direct product) of line segments and circles. This allows the spectral basis functions to be the product of one-dimensional factors and the computational grid to be a tensor product of one-dimensional grids. Through partial summation, multidimensional sums and interpolations can be performed as a sequence of one-dimensional sums in interpolations had a huge saving in cost relative to the expense when there is no tensor product structure to be exploited. Furthermore, the multidimensional tensor product methods are built on the firm and precise foundation of one-dimensional convergence theory and one-dimensional canonical grids and basis functions. Unfortunately, many interesting problems are posed and complicated geometry. The ocean, for example, is full of islands, bays, peninsulas, and estuaries. Multi-domain methods work well, but require complicated bookkeeping. A single domain approach in which a single series is the global approximation Is much simpler. Radial basis functions are a so-called "meshless" method which is tolerant of complicated geometry and does not require that the domain be subdivided into many subdomains. However, there are many significant foundational questions about radial basis functions such as the choice of numerical parameters, the optimum distribution of grid points, rates of convergence, vulnerability to the Runge Phenomena, and relationship to non-tensor polynomial algorithms. We review the problem and potential of spectral methods in complex geometry.