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The radial basis function-differential quadrature method for elliptic problems in annular domains. (English) Zbl 1418.65188
Summary: We employ a radial basis function (RBF)-differential quadrature (DQ) method for the numerical solution of elliptic boundary value problems in annular domains. With an appropriate selection of collocation points, for any choice of RBF, both the coefficient and right hand side matrices in the systems appearing in this discretization possess block circulant structures. These linear systems can thus be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy-Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
Software:
Matlab
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[1] Bai, Y.; Wu, Y.; Xie, X., Uniform convergence analysis of a higher order hybrid stress quadrilateral finite element method for linear elasticity, Adv. Appl. Math. Mech., 8, 3, 399-425, (2016)
[2] Bai, Y. H.; Wu, Y. K.; Xie, X. P., Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method, Sci. China Math., 59, 1835-1850, (2016) · Zbl 1388.74091
[3] Wu, H. Y.; Duan, Y., Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis-Procesi equation, Appl. Math. Comput., 274, 83-92, (2016) · Zbl 1410.65331
[4] Bellman, R.; Kashef, B. G.; Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys., 10, 40-52, (1972) · Zbl 0247.65061
[5] Wu, Y. L.; Shu, C., Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli, Comput. Mech., 29, 477-485, (2002) · Zbl 1146.76635
[6] Ding, H.; Shu, C.; Yeo, K. S.; Lu, Z. L., Simulation of natural convection in eccentric annuli between a square outer cylinder and a circular inner cylinder using local MQ-DQ method, Numer. Heat Transfer A, 47, 291-313, (2005)
[7] Hidayat, M. I.P.; Ariwahjoedi, B.; Parman, S., A new meshless local B-spline basis functions-FD method for two-dimensional heat conduction problems, Internat. J. Numer. Methods Heat Fluid Flow, 25, 225-251, (2015) · Zbl 1356.80064
[8] Homayoon, L.; Abadini, M. J.; Hashemi, S. M.R., RBF-DQ solution for shallow water equations, J. Waterway Port Coastal Ocean Eng., 139, 45-60, (2013)
[9] Korkmaz, A.; Dağ, I., Solitary wave simulations of complex modified Korteweg – de Vries equation using differential quadrature method, Comput. Phys. Comm., 180, 1516-1523, (2009)
[10] Krowiak, A., Hermite type radial basis function-based differential quadrature method for higher order equations, Appl. Math. Model., 40, 2421-2430, (2016)
[11] Shen, L. H.; Tseng, K. H.; Young, D. L., Evaluation of multi-order derivatives by local radial basis function differential quadrature method, J. Mech., 29, 67-78, (2013)
[12] Shu, C.; Wu, Y. L., Integrated radial basis functions-based differential quadrature method and its performance, Internat. J. Numer. Methods Fluids, 53, 969-984, (2007) · Zbl 1109.65025
[13] Wu, Y. L.; Shu, C.; Chen, H. Q.; Zhao, N., Radial basis function enhanced domain-free discretization method and its applications, Numer. Heat Transfer B, 46, 269-282, (2004)
[14] Bialecki, B.; Fairweather, G.; Karageorghis, A., Matrix decomposition algorithms for elliptic boundary value problems: A survey, Numer. Algorithms, 56, 253-295, (2011) · Zbl 1208.65036
[15] Liu, X. Y.; Karageorghis, A.; Chen, C. S., A Kansa-radial basis function method for elliptic boundary value problems in annular domains, J. Sci. Comput., 65, 1240-1269, (2015) · Zbl 1328.65254
[16] Karageorghis, A.; Chen, C. S.; Liu, X.-Y., Kansa-RBF algorithms for elliptic problems in axisymmetric domains, SIAM J. Sci. Comput., 38, A435-A470, (2016) · Zbl 1432.65174
[17] Chen, C. S.; Karageorghis, A., Local RBF algorithms for elliptic boundary value problems in annular domains, Commun. Comput. Phys., 25, 41-67, (2019)
[18] Karageorghis, A.; Chen, C. S.; Smyrlis, Y.-S., A matrix decomposition RBF algorithm: approximation of functions and their derivatives, Appl. Numer. Math., 57, 304-319, (2007) · Zbl 1107.65305
[19] Karageorghis, A.; Chen, C. S.; Smyrlis, Y.-S., Matrix decomposition RBF algorithm for solving 3D elliptic problems, Eng. Anal. Bound. Elem., 33, 1368-1373, (2009) · Zbl 1244.65184
[20] Davis, P. J., Circulant Matrices, (1994), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, Rhode Island · Zbl 0898.15021
[21] Heryudono, A. R.H.; Driscoll, T. A., Radial basis function interpolation on irregular domain through conformal transplantation, J. Sci. Comput., 44, 286-300, (2010) · Zbl 1203.65025
[22] . The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA, Matlab.
[23] Hartmann, F., Elastostatics, Progress in Boundary Element Methods. Vol. 1, 84-167, (1981), Pentech Press: Pentech Press London
[24] Rippa, S., An algorithm for selecting a good value for the parameter \(c\) in radial basis function interpolation, Adv. Comput. Math., 11, 193-210, (1999) · Zbl 0943.65017
[25] Fasshauer, G. E.; Zhang, J. G., On choosing optimal shape parameters for RBF approximation, Numer. Algorithms, 45, 345-368, (2007) · Zbl 1127.65009
[26] http://www.math.usm.edu/cschen/JCAM/Example1.m,Example2.m,Example3.m.
[27] Buhmann, M. D., A new class of radial basis functions with compact support, Math. Comp., 70, 307-318, (2001) · Zbl 0956.41002
[28] Li, M.; Chen, W.; Chen, C. S., The localized RBFs collocation methods for solving high dimensional PDEs, Eng. Anal. Bound. Elem., 37, 1300-1304, (2013) · Zbl 1287.65115
[29] Karageorghis, A., The method of fundamental solutions for elliptic problems in circular domains with mixed boundary conditions, Numer. Algorithms, 68, 185-211, (2015) · Zbl 1308.65210
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