Interdisciplinary Applied Mathematics

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FIGURE 14.8. Four snapshots for mixing promoted by an oscillating cylinder. The concentration density contours are shown for Re = 100, Sc = 5,St = 0.6 A total of 346 spectral/hp elements with 8th order modal expansion are used in the simulations. Arrows show the direction of motion of the cylinder.

14.2 Meshless Methods

The spectral element method presented above is a typical Galerkin method, and in the limit of linear basis it reduces to the standard finite element method. The high-order equivalent of the finite volume method, which would be more appropriate for compressible high-speed flows, e.g., in micronozzles, is a discontinuous Galerkin method with a spectral basis, see (Cockburn et al., 2000), and references therein. Both classes of methods and their low-order counterparts are based on a mesh that consists of triangles, quadrilaterals in two dimensions, and tetrahedra, hexahedra, etc. in three dimensions.

A popular research topic in numerical methods has been the development of meshless methods as alternatives to the traditional finite element, finite volume, and finite difference methods. The traditional methods all require some connectivity knowledge a priori, such as the generation of a mesh,

whereas the aim of meshless methods is to sprinkle only a set of points or nodes covering the computational domain, with no connectivity information required among the set of points. Multiphysics (specifically involving problems with moving domains) and multiscale analysis can be simplified by meshless techniques,    since    we    deal    with    only    nodes    or    points    instead

of a mesh. Meshless techniques are also appealing because of their potential in adaptive techniques, where a user can simply add more points in a particular region to obtain more accurate results.

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