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ABSTRACT
In this article, we propose a new two-level method for solving 2D and 3D semilinear elliptic boundary value problem base on radial basis function (RBF). In the first step, we use the global RBF method to solve a semilinear problem on the coarse mesh or a small number of collocation points, in the second step, finite difference (FD), finite element (FE), and radial basis function-generated finite difference (RBF-FD) methods are used to solve the linearized problem on the fine mesh or a large number of collocation points, respectively. Numerical examples are provided to verify the feasibility and efficiency of the two-level RBF method. Moreover, compared to the FD and FE methods, RBF-FD method used in the second step further improve the accuracy of numerical solution.
Nomenclature
| Ω | = | a bounded domain |
| ∂Ω | = | boundary of the domain |
| x | = | space variable |
| xi | = | center point |
| u | = | unknown function |
| uH | = | solution on the coarse mesh |
| uh | = | solution on the fine mesh |
| f | = | function of x and u |
| g | = | source function |
| ℝ | = | real number set |
| N | = | number of collocation points |
| NI | = | number of the interior nodes |
| ni | = | number of the neighborhood of xi |
| λj | = | interpolation coefficient |
| ϕ | = | radial basis function |
| Δ | = | Laplacian operator |
| γ | = | constant |
| wim | = | weight coefficient |
| ∥ · ∥ | = | Euclidean distance function |
| ∥ · ∥∞ | = | infinite norm |
| c | = | shape parameter |
| cH | = | shape parameter on the coarse mesh |
| ch | = | shape parameter on the fine mesh |
| A, B | = | coefficient matrix |
| Λ, G, U | = | column vector |
| Subscripts | = | |
| i | = | ith component |
| j | = | jth component |
| m | = | mth component |
| Superscripts | = | |
| s | = | iteration number |
| d | = | dimension of ℝ |
Nomenclature
| Ω | = | a bounded domain |
| ∂Ω | = | boundary of the domain |
| x | = | space variable |
| xi | = | center point |
| u | = | unknown function |
| uH | = | solution on the coarse mesh |
| uh | = | solution on the fine mesh |
| f | = | function of x and u |
| g | = | source function |
| ℝ | = | real number set |
| N | = | number of collocation points |
| NI | = | number of the interior nodes |
| ni | = | number of the neighborhood of xi |
| λj | = | interpolation coefficient |
| ϕ | = | radial basis function |
| Δ | = | Laplacian operator |
| γ | = | constant |
| wim | = | weight coefficient |
| ∥ · ∥ | = | Euclidean distance function |
| ∥ · ∥∞ | = | infinite norm |
| c | = | shape parameter |
| cH | = | shape parameter on the coarse mesh |
| ch | = | shape parameter on the fine mesh |
| A, B | = | coefficient matrix |
| Λ, G, U | = | column vector |
| Subscripts | = | |
| i | = | ith component |
| j | = | jth component |
| m | = | mth component |
| Superscripts | = | |
| s | = | iteration number |
| d | = | dimension of ℝ |
Acknowledgments
The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this article.