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

无网格方法是一种现代的替代经典的基于网格的方法离散偏微分方程。与这些经典的方法相比，它们不需要耗时的建立计算网格的步骤。在这个项目中，一个新的，高阶无网格方法离散偏微分方程将被开发，分析和实施。特别强调将在经典的流体动力学微分方程。所提出的方法在其方法上是独特的。它采用径向基函数，在以下方面与经典方案显着不同。对于不可压缩流体，它基于解析发散自由近似空间，并创建速度和压力的连续近似，这样既不需要分裂技术也不需要inf-sup条件，甚至不需要额外的计算。此外，它不需要任何形式的数值积分，并灵活地处理与时间有关的变化的基础几何。除了严格的数学分析，它将彻底分析是否和以何种方式可以减少计算复杂性可以实现。此外，它的目的是表明，这种离散化方法提供了一种新的，简单的方法和一个显着的改进，用于分析和提取流相关的功能。虽然主要的重点是从流体力学的应用，所提出的方法，经过一些修改，可以适用于其他领域，如金融期权价格的建模和生命科学中的癌症生长的建模。新的主要课题，这一延续应用的数学误差和稳定性分析的一般边界数据的扩展，条件正定核的相应理论的发展以及有效数值算法的发展、实现和分析。

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.