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[1] Fürst, J. & Sonar, T., On meshless collocation approximations of conservation laws: pre-liminary investigations on positive schemes and dissipation models. ZAMMZeitschrift fïir Angewandte Mathematik und Mechanik Journal of Applied Mathematics and Mechanics, 81(6), pp. 403–415, 2001.
[2] Zhou, X., Hon, Y. & Cheung, K., A grid-free, nonlinear shallow-water model with mov-ing boundary. Engineering Analysis with Boundary Elements, 28(9), pp. 1135–1147, 2004. [Crossref]
[3] Kansa, E.J. & Geiser, J., Numerical solution to time-dependent 4D invis-cid Burgers’ equations. Engineering Analysis with Boundary Elements, 37(3), pp. 637–645, 2013. [Crossref]
[4] Li, W., Li, M., Chen, C.S. & Liu, X., Compactly supported radial basis functions for solving certain high order partial differential equations in 3D. Engineering Analysis with Boundary Elements, 55(SI), pp. 2–9, 2015.
[5] Pang, G., Chen, W. & Fu, Z., Space-fractional advection-dispersion equations by the Kansa method. Journal of Computational Physics, 293(SI), pp. 280–296, 2015.
[6] Kansa, E.J., Multiquadricsa scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative esti-mates. Computers & Mathematics with applications, 19(8), pp. 127–145,1990. [Crossref]
[7] Kansa, E.J., Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers & Mathematics with Applications, 19(8–9), pp. 147–161, 1990. [Crossref]
[8] Hon, Y. & Schaback, R., On unsymmetric collocation by radial basis functions. Applied Mathematics and Computation, 119(2), pp. 177–186, 2001.
[9] Fasshauer, G.E., Solving differential equations with radial basis functions: multi-level methods and smoothing. Advances in Computational Mathematics, 11(2–3), pp. 139–159,1999.
[10] Schaback, R., Convergence of unsymmetric kernel-based meshless collocation methods. SIAM Journal on Numerical Analysis, 45(1), pp. 333–351, 2007.
[11] Cheung, K.C., Ling, L. & Schaback, R., H2-convergence of least squares kernel collocation method, Submitted 2016.
[12] Hu, H., Chen, J. & Hu, W., Weighted radial basis collocation method for boundary value problems. International Journal for Numerical Methods in Engineering, 69(13), pp. 2736–2757, 2007.
[13] Ling, L. & Schaback, R., Stable and convergent unsymmetric meshless collocation methods. SIAM Journal on Numerical Analysis, 46(3), pp. 1097–1115, 2008.
[14] Ling, L. & Schaback, R., An improved subspace selection algorithm for meshless collo-cation methods. International Journal for Numerical Methods in Engineering, 80(13), pp. 1623–1639, 2009.
[15] Ling, L., A fast block-greedy algorithm for quasi-optimal meshless trial sub-space selection. SIAM Journal on Scientific Computing, 38(2), pp. A1224–A1250, 2016.
[16] Ling, L., available at: http://www.math.hkbu.edu.hk/~lling/blockgreedy.m.
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Open Access
Research article

Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method

ka chun cheung,
leevan ling
Department of Mathematics, Hong Kong Baptist University
International Journal of Computational Methods and Experimental Measurements
|
Volume 5, Issue 3, 2017
|
Pages 377-386
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: 03-31-2017
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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 K columns (or trial centers) out of an M × N 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
Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References
[1] Fürst, J. & Sonar, T., On meshless collocation approximations of conservation laws: pre-liminary investigations on positive schemes and dissipation models. ZAMMZeitschrift fïir Angewandte Mathematik und Mechanik Journal of Applied Mathematics and Mechanics, 81(6), pp. 403–415, 2001.
[2] Zhou, X., Hon, Y. & Cheung, K., A grid-free, nonlinear shallow-water model with mov-ing boundary. Engineering Analysis with Boundary Elements, 28(9), pp. 1135–1147, 2004. [Crossref]
[3] Kansa, E.J. & Geiser, J., Numerical solution to time-dependent 4D invis-cid Burgers’ equations. Engineering Analysis with Boundary Elements, 37(3), pp. 637–645, 2013. [Crossref]
[4] Li, W., Li, M., Chen, C.S. & Liu, X., Compactly supported radial basis functions for solving certain high order partial differential equations in 3D. Engineering Analysis with Boundary Elements, 55(SI), pp. 2–9, 2015.
[5] Pang, G., Chen, W. & Fu, Z., Space-fractional advection-dispersion equations by the Kansa method. Journal of Computational Physics, 293(SI), pp. 280–296, 2015.
[6] Kansa, E.J., Multiquadricsa scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative esti-mates. Computers & Mathematics with applications, 19(8), pp. 127–145,1990. [Crossref]
[7] Kansa, E.J., Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers & Mathematics with Applications, 19(8–9), pp. 147–161, 1990. [Crossref]
[8] Hon, Y. & Schaback, R., On unsymmetric collocation by radial basis functions. Applied Mathematics and Computation, 119(2), pp. 177–186, 2001.
[9] Fasshauer, G.E., Solving differential equations with radial basis functions: multi-level methods and smoothing. Advances in Computational Mathematics, 11(2–3), pp. 139–159,1999.
[10] Schaback, R., Convergence of unsymmetric kernel-based meshless collocation methods. SIAM Journal on Numerical Analysis, 45(1), pp. 333–351, 2007.
[11] Cheung, K.C., Ling, L. & Schaback, R., H2-convergence of least squares kernel collocation method, Submitted 2016.
[12] Hu, H., Chen, J. & Hu, W., Weighted radial basis collocation method for boundary value problems. International Journal for Numerical Methods in Engineering, 69(13), pp. 2736–2757, 2007.
[13] Ling, L. & Schaback, R., Stable and convergent unsymmetric meshless collocation methods. SIAM Journal on Numerical Analysis, 46(3), pp. 1097–1115, 2008.
[14] Ling, L. & Schaback, R., An improved subspace selection algorithm for meshless collo-cation methods. International Journal for Numerical Methods in Engineering, 80(13), pp. 1623–1639, 2009.
[15] Ling, L., A fast block-greedy algorithm for quasi-optimal meshless trial sub-space selection. SIAM Journal on Scientific Computing, 38(2), pp. A1224–A1250, 2016.
[16] Ling, L., available at: http://www.math.hkbu.edu.hk/~lling/blockgreedy.m.
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Cheung, K. C. & Ling, L. (2017). Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method. Int. J. Comput. Methods Exp. Meas., 5(3), 377-386. https://doi.org/10.2495/CMEM-V5-N3-377-386
K. C. Cheung and L. Ling, "Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method," Int. J. Comput. Methods Exp. Meas., vol. 5, no. 3, pp. 377-386, 2017. https://doi.org/10.2495/CMEM-V5-N3-377-386
@research-article{Cheung2017ConvergenceSF,
title={Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method},
author={Ka Chun Cheung and Leevan Ling},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2017},
page={377-386},
doi={https://doi.org/10.2495/CMEM-V5-N3-377-386}
}
Ka Chun Cheung, et al. "Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method." International Journal of Computational Methods and Experimental Measurements, v 5, pp 377-386. doi: https://doi.org/10.2495/CMEM-V5-N3-377-386
Ka Chun Cheung and Leevan Ling. "Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method." International Journal of Computational Methods and Experimental Measurements, 5, (2017): 377-386. doi: https://doi.org/10.2495/CMEM-V5-N3-377-386
CHEUNG K C， LING L. Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method[J]. International Journal of Computational Methods and Experimental Measurements， 2017， 5(3)： 377-386. https://doi.org/10.2495/CMEM-V5-N3-377-386