Meshless local Petrov Galerkin method for 2D/3D nonlinear convection–diffusion equations based on LS-RBF-PUM

References

• B. Guo, Difference Method of Partial Differential Equation. Beijing, China: Science Press, 1988.
   Google Scholar
• B. Du, H. Su, and X. Feng, “Two-level variational multiscale method based on the decoupling approach for the natural convection problem,” Int. Comm. Heat and Mass Trans., vol. 61, pp. 128–139, 2015.
   Web of Science ®Google Scholar
• P. Huang, J. Zhao, and X. Feng, “Highly efficient and local projection-based stabilized finite element method for natural convection problem,” Int. J. Heat and Mass Trans., vol. 83, pp. 357–365, 2015.
   Web of Science ®Google Scholar
• J. Wu, P. Huang, and X. Feng, “A new variational multiscale FEM for the steady-state natural convection problem with bubble stabilization,” Num. Heat Trans., Part A: App., vol. 68, no. 7, pp. 777–796, 2015.
   Web of Science ®Google Scholar
• J. Wu, P. Huang, X. Feng, and D. Liu, “An efficient two-step algorithm for steady-state natural convection problem,” Int. J. Heat and Mass Trans., vol. 101, pp. 387–398, 2016.
   Web of Science ®Google Scholar
• T. Zhang, X. Feng, and J. Yuan, “Implicit-explicit schemes of finite element method for the non-stationary thermal convection problems with temperature-dependent coefficients,” Int. Comm. Heat and Mass Trans., vol. 76, pp. 325–336, 2016.
   Web of Science ®Google Scholar
• L. Qian, H. Cai, R. Guo, and X. Feng, “The characteristic variational multiscale method for convection dominated convection–diffusion-reaction problems,” Int. J. Heat and Mass Trans., vol. 72, pp. 461–469, 2014.
   Web of Science ®Google Scholar
• Z. Si, X. Feng, and A. Abduwali, “The semi-discrete streamline diffusion finite element method for timedependented convection–diffusion problems,” App. Math. Comp., vol. 202, no. 2, pp. 771–779, 2008.
   Web of Science ®Google Scholar
• W. Wu, X. Feng, and D. Liu, “The local discontinuous galerkin finite element method for a class of convection–diffusion equations,” Nonlinear Anal.: Real World App., vol. 14, no. 1, pp. 734–752, 2013.
   Web of Science ®Google Scholar
• N. Li, J. Zhao, X. Feng, and D. Gui, “Generalized polynomial chaos for the convection diffusion equation with uncertainty,” Int. J. Heat and Mass Trans., vol. 97, pp. 289–300, 2016.
   Web of Science ®Google Scholar
• L. Qian, X. Feng, and Y. He, “The characteristic finite difference streamline diffusion method for convectiondominated diffusion problems,” App. Math. Model., vol. 36, no. 2, pp. 561–572, 2012.
   Web of Science ®Google Scholar
• S. Zhai, X. Feng, and D. Liu, “A novel method to deduce a high-order compact difference scheme for the three-dimensional semilinear convection–diffusion equation with variable coefficients,” Num. Heat Trans., Part B: Fund., vol. 63, no. 5, pp. 425–455, 2013.
   Web of Science ®Google Scholar
• S. Zhai, X. Feng, and Y. He, “A new high-order compact ADI method for 3-D unsteady convection–diffusion problems with discontinuous coefficients,” Num. Heat Trans., Part B: Fund., vol. 65, no. 4, pp. 376–391, 2014.
   Web of Science ®Google Scholar
• S. Zhai, X. Feng, and Y. He, “An unconditionally stable compact ADI method for threedimensional timefractional convection–diffusion equation,” J. Comp. Phy., vol. 269, pp. 138–155, 2014.
   Web of Science ®Google Scholar
• S. Zhai, X. Feng, and Z. Weng, “New high-order compact adi algorithms for 3D nonlinear time-fractional convection–diffusion equation,” Math. Prob. Eng., vol. 2013, pp. 1, 2013.
   Google Scholar
• S. Zhai, D. Gui, P. Huang, and X. Feng, “A novel high-order ADI method for 3D fractional convectiondiffusion equations,” Int. Comm. Heat and Mass Trans., vol. 66, pp. 212–217, 2015.
   Web of Science ®Google Scholar
• S. Zhai, L. Qian, D. Gui, and X. Feng, “A block-centered characteristic finite difference method for convection-dominated diffusion equation,” Int. Comm. Heat and Mass Trans., vol. 61, pp. 1–7, 2015.
   Web of Science ®Google Scholar
• J. Li, S. Zhai, Z. Weng, and X. Feng, “H-adaptive RBF-FD method for the high-dimensional convection–diffusion equations,” Int. Comm. Heat and Mass Trans., vol. 89, pp. 139–146, 2017.
   Web of Science ®Google Scholar
• N. Li, H. Su, D. Gui, and X. Feng, “Multiquadric RBF-FD method for the convection-dominated diffusion problems base on shishkin nodes,” Int. J. Heat and Mass Trans., vol. 118, pp. 734–745, 2018.
   Web of Science ®Google Scholar
• Y. Qiao, S. Zhai, and X. Feng, “RBF-FD method for the high dimensional time fractional convection–diffusion equations,” Int. Comm. Heat and Mass Trans., vol. 89, pp. 230–240, 2017.
   Web of Science ®Google Scholar
• A. Shirzadi, L. Ling, and S. Abbasbandy, “Meshless simulations of the two-dimensional fractional-time convection–diffusion-reaction equations,” Eng. Anal. with Boundary Ele., vol. 36, no. 11, pp. 1522–1527, 2012.
   Web of Science ®Google Scholar
• Z. Chen, Z. Li, W. Xie, and X. Wu, “A Two-Level Variational Multiscale Meshless Local Petrov-Galerkin (VMS-MLPG) Method for Convection-Diffusion Problems with Large Peclet Number,” Comput. Fluids, vol. 164, pp. 73–82, 2018.
   Google Scholar
• D. Hu, S. Long, K. Liu, and G. Li, “A modified meshless local Petrov-Galerkin method to elasticity problems in computer modeling and simulation,” Eng. Anal. with Boundary Ele., vol. 30, no. 5, pp. 399–404, 2006.
   Web of Science ®Google Scholar
• K. Liu, S. Long, and G. Li, “A simple and less-costly meshless local Petrov-Galerkin (MLPG) method for the dynamic fracture problem,” Eng. Anal. with Boundary Ele., vol. 30, no. 1, pp. 72–76, 2006.
   Web of Science ®Google Scholar
• S. N. Atluri, The Meshless Method (MLPG) for Domain & Bie Discretizations, and Encino, CA: Tech Science Press, 2004.
   Google Scholar
• I. Babuška, and J. M. Melenk, “The partition of Unity method,” Int. J. Numer. Meth. Eng., vol. 40, no. 4, pp. 727–758, 1997.
   Web of Science ®Google Scholar
• H. Wendland, “Fast evaluation of radial basis functions: Methods based on partition of Unity,” Wave. Splines & App., pp. 473–483, 2002.
   Google Scholar
• G. F. Fasshauer, “Meshfree approximation methods with MATLAB,” World Scientific, 2007.
   Google Scholar
• R. Cavoretto, “Partition of Unity algorithm for two-dimensional interpolation using compactly supported radial basis functions,” Appl. Math. Inf. Sci, vol. 9, no. 1, pp. 1–13, 2015.
   Google Scholar
• R. Cavoretto, A. D. Rossi, and E. Perracchione, RBF-PUM Interpolation with Variable Subdomain Sizes and Shape Parameters//American Institute of Physics Conference Series. AIP Publishing LLC, 2016.
   Google Scholar
• S. De Marchi, A. Martnez, E. Perracchione, and M. Rossini, “RBF-based partition of Unity method for elliptic PDEs: Adaptivity and stability issues via VSKs,” Submitted to Journal of Scientific Computing, 2017.
   Google Scholar
• A. Safdari-Vaighani, A. Heryudono, and E. Larsson, “A radial basis function partition of Unity collocation method for convection–diffusion equations arising in financial applications,” J. Sci. Comp., vol. 64, no. 2, pp. 341–367, 2015.
   Web of Science ®Google Scholar
• E. Larsson, V. Shcherbakov, and A. Heryudono, “A least square radial basis function partition of Unity methods for solving PDEs,” SIAM J. Sci. Comp., vol. 39, no. 6, pp. A2538–A2563, 2017.
   Web of Science ®Google Scholar
• N. Li, Z. Tan, and X. Feng, “Novel two-level discretization method for high dimensional semilinear elliptic problems base on RBF-FD scheme,” Numer. Heat Trans., Part B: Fund., vol. 72, no. 5, pp. 349–360, 2017.
   Web of Science ®Google Scholar
• V. Bayona, M. Moscoso, and M. Kindelan, “Optimal constant shape parameter for multiquadric based RBF-FD method,” J. Comp. Phy., vol. 230, no. 19, pp. 7384–7399, 2011.
   Web of Science ®Google Scholar
• V. Bayona, M. Moscoso, M. Carretero, and M. Kindelan, “RBF-FD formulas and convergence properties,” J. Comp. Phy., vol. 229, no. 22, pp. 8281–8295, 2010.
   Web of Science ®Google Scholar
• C. A. Micchelli, Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions//Approximation Theory and Spline Functions. Netherlands: Springer, 1984.
   Google Scholar
• D. Shepard, “A two-dimensional interpolation function for irregularly-spaced data,” Proc. ACM Nat. Conf., pp. 517–524, 1968.
   Google Scholar
• H. Wendland, Scattered Data Approximation. Cambridge University Press, 2004.
   Google Scholar
• W. R. Madych, “Miscellaneous error bounds for multiquadric and related interpolators,” Comp. & Math. with App., vol. 24, no. 12, pp. 121–138, 1992.
   Web of Science ®Google Scholar