Mesh generation is still the most time consuming part of any mesh-based numerical simulation. Another obvious advantage of meshfree discretizations is - of course - their independence from a mesh. Meshfree methods are often better suited to cope with changes in the geometry of the domain of interest (e.g., free surfaces and large deformations) than classical discretization techniques such as finite differences, finite elements or finite volumes. Many traditional numerical methods can either not handle such problems at all, or are limited to very special (regular) situations. Moreover, computation with high-dimensional data is an important issue in many areas of science and engineering. Thus, much of the work concerned with meshfree approximation methods is interdisciplinary - at the interface between mathematics and numerous application areas (see the partial list below). Meshfree methods have gained much attention in recent years, not only in the mathematics but also in the engineering community.
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