Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method

Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method

Ka Chun Cheung Leevan Ling

Department of Mathematics, Hong Kong Baptist University

Page:
377-386
|
DOI:
https://doi.org/10.2495/CMEM-V5-N3-377-386
Received:
N/A
| |
Accepted:
N/A
| | Citation

OPEN ACCESS

Abstract:

In this paper, we apply the recently proposed fast block-greedy algorithm to a convergent kernel-based collocation method. In particular, we discretize three-dimensional second-order elliptic differential equations by the meshless asymmetric collocation method with over-sampling. Approximated solutions are obtained by solving the resulting weighted least squares problem. Such formulation has been proven to have optimal convergence in H2. Our aim is to investigate the convergence behaviour of some three dimensional test problems. We also study the low-rank solution by restricting the approximation in some smaller trial subspaces. A block-greedy algorithm, which costs at most O(NK2) to select columns (or trial centers) out of an × overdetermined matrix, is employed for such an adaptivity. Numerical simulations are provided to justify these reductions.

Keywords:

ansa method, kernel-based collocation, adaptive greedy algorithm, elliptic equation

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