Meshfree methods for the numerical solution of partial differential equations from fluid dynamics

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Meshfree methods represent a modern alternative to classsical mesh-based methods for discretising partial differential equations. In contrast to these classical methods, they do not require the time-consuming step of building a computational mesh.In this project, a novel, high-order meshfree method for discretising partial differential equations will be developed, analysed and implemented. Special emphasis will be on classical differential equations from fluid dynamics. The proposed method is unique in its approach. It employs radial basis functions and differs significantly from classical schemes in the following ways. For incompressible fluids, it is based upon analytically divergence free approximation spaces and creates simultanous approximations for the velocity and pressure such that neither a splitting technique nor an inf-sup condition, not even an additional computation, are required. Moreover, it does not require any kind of numericalintegration and is flexible in handling time-dependent changes of theunderlying geometry.Besides a rigorous mathematical analysis, it will be thoroughly analysed whether and in which way a reduction in computational complexity can be achieved. Furthermore, it is intended to show that this discretisation method provides a new, simple approach and a significant improvement for analysing and extracting flow relevant features. Though the main focus is on applications from fluid dynamics, the proposed method can, after some modifications, be applied to other areas such as the modelling of option prices in finance and the modelling of cancer growth in the life sciences.The new main topics of this continuation application are the extension of the mathematical error and stability analysis to general boundary data, the development of a corresponding theory for conditionally positive definite kernels and the development, the implementation and the analysis of efficient numerical algorithms.