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则性来减小模板尺寸，从而降低无网格近似总计算成本的方法。与在整个计算域内布设散乱节点不同，仅在具有几何细节的区域布设散乱节点，而其余区域则使用规则节点进行离散化。因此，在规则节点覆盖的区域，仅使用单项式基函数即可进行近似计算，从而有效减小了所需模板尺寸；而在散乱节点区域，则需要添加一组多调和样条函数以确保收敛行为。本文以自然对流驱动的流体流动问题为例，研究了所提出的混合散乱-规则近似方法在计算效率与数值解精度两方面的性能。我们从定义于方形域上的de Vahl Davis基准算例开始求解，进而拓展至二维与三维不规则形状域。研究表明，两种近似方法的空间变化可在仅对解精度产生微小影响的前提下显著降低计算复杂度。