基于点插值的配点型无网格法解Helmholtz问题

基于点插值法的思想,用三角函数作为基函数在局部支持域内构造具有Kroneckerδ函数性、单位分解性、高阶连续性、再生性和紧支性的形函数.用配点法离散微分方程,得到了具有稀疏带状性的系数矩阵,用GMERS方法求解代数方程组,分别研究了Helmholtz问题的边界层问题和波传播问题.通过数值算例可以发现,给出的数值结果非常接近于精确解,且随着节点的增加,其精确度越来越高,具有良好的收敛性.

Solving Helmholtz problem by collocation meshless method based on point interpolation

Combining the point interpolation method with trigonometric functions which are used as base functions, a shape function is structured in the local support domain. The shape function has many properties, such as Kronecker functionality, unit decomposition and reproducibility as well as compact properties. Discreting differential equations by the allocation method, a sparse band coefficient matrix is obtained. The GMERS method is used to solve algebraic equations. Two kinds of Helmholtz problem: a boundary layer problem and a wave propagation problem are solved. Numerical examples can be found, and the results are close to the exact solutions. Furthermore, high precision and good convergence could be obtained as the nodes increased.