In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatial variation of computational node regularity. Rather than covering the entire domain with scattered nodes, only regions with geometric details are covered with scattered nodes, while the rest of the domain is discretised with regular nodes. Consequently, in regions covered with regular nodes the approximation using solely the monomial basis can be performed, effectively reducing the required stencil size compared to the approximation on scattered nodes where a set of polyharmonic splines is added to ensure convergent behaviour. The performance of the proposed hybrid scattered-regular approximation approach, in terms of computational efficiency and accuracy of the numerical solution, is studied on natural convection driven fluid flow problems. We start with the solution of the de Vahl Davis benchmark case, defined on square domain, and continue with two- and three-dimensional irregularly shaped domains. We show that the spatial variation of the two approximation methods can significantly reduce the computational complexity, with only a minor impact on the solution accuracy.

翻译：本文提出了一种通过空间变化计算节点规律性来减小模板尺寸，从而降低无网格逼近总计算成本的方法。该方法并非在整个域中散布节点，仅在具有几何细节的区域使用散布节点，而其余区域则采用规则节点离散化。因此，在规则节点覆盖的区域，可仅使用单项式基进行逼近，有效减小了模板尺寸（相较于散布节点逼近，后者需添加多调和样条以确保收敛性）。我们以自然对流驱动流体流动问题为例，研究了所提出的混合散布-规则逼近方法在计算效率和数值解精度方面的性能。首先求解定义在方形域上的de Vahl Davis基准案例，继而扩展至二维和三维不规则形状域。结果表明，两种逼近方法的空间变化可显著降低计算复杂度，且对解精度影响极小。