我們提出了一種非嵌套的多級算法，可以使用多諧波徑向基函數（PHS-RBF）插值來求解離散點處離散的Poisson方程。我們将多項式附加到徑向基函數上，以實作離散化誤差的指數收斂。插值是在點的局部雲上執行的，泊松方程在每個分散的點處并置，進而導緻未知變量的稀疏離散方程組。為了解決這組方程，我們開發了一種非嵌套的多級算法，該算法利用多個獨立生成的粗糙點集。限制和延長算子也使用相同的RBF插值過程構造。使用制造的解決方案在三種模型幾何中評估Dirichlet和全Neumann邊界條件算法的性能。對于Dirichlet邊界條件，使用SOR點求解器作為松弛方案可以觀察到快速收斂。對于全Neumann邊界條件，收斂性随附加多項式的程度而變慢。但是，當将多級過程與GMRES算法結合使用時，收斂性會得到顯着改善。分數步法中包含了GMRES加速多級算法，用于求解不可壓縮的Navier-Stokes方程。
原文題目：A Non-Nested Multilevel Method for Meshless Solution of the Poisson Equation in Heat Transfer and Fluid Flow
原文：We present a non-nested multilevel algorithm for solving the Poisson equation discretized at scattered points using polyharmonic radial basis function (PHS-RBF) interpolations. We append polynomials to the radial basis functions to achieve exponential convergence of discretization errors. The interpolations are performed over local clouds of points and the Poisson equation is collocated at each of the scattered points, resulting in a sparse set of discrete equations for the unkown variables. To solve this set of equations, we have developed a non-nested multilevel algorithm utilizing multiple independently generated coarse sets of points. The restriction and prolongation operators are also constructed with the same RBF interpolations procedure. The performance of the algorithm for Dirichlet and all-Neumann boundary conditions is evaluated in three model geometries using a manufactured solution. For Dirichlet boundary conditions, rapid convergence is observed using SOR point solver as the relaxation scheme. For cases of all-Neumann boundary conditions, convergence is seen to slow down with the degree of the appended polynomial. However, when the multilevel procedure is combined with a GMRES algorithm, the convergence is seen to significantly improve. The GMRES accelerated multilevel algorithm is included in a fractional step method to solve incompressible Navier-Stokes equations.
傳熱和流體流動泊松方程無網格求解的非嵌套多級方法.pdf