A meshless local Petrov–Galerkin approach for solving the convection-dominated problems based on the streamline upwind idea and the variational multiscale concept

References

• X. Zhang, J. Ouyang, and J. Wang, “Stabilization meshless method for convection-dominated problems,” Appl. Math. Mech., vol. 29, pp. 1065–1075, 2008. DOI: 10.1007/s10483-008-0810-y.
   Google Scholar
• S. Khankham, A. Luadsong, and N. Aschariyaphotha, “MLPG method based on moving kriging interpolation for solving convection–diffusion equations with integral condition,” J. King Saud Sci., vol. 34, pp. 1018–1028, 2015. DOI: 10.1016/j.jksus.2015.03.001.
   Google Scholar
• H. Lin and S. N. Atluri, “Meshless local Petrov–Galerkin (MLPG) method for convection–diffusion problems,” CMES Comput. Model. Eng. Sci., vol. 1, pp. 45–60, 2000.
   Web of Science ®Google Scholar
• H. Lin and S. N. Atluri, “The meshless local Petrov–Galerkin (MLPG) method for solving incompressible Navier-Stokes equation,” CMES Comput. Model. Eng. Sci., vol. 2, pp. 117–142, 2001.
   Web of Science ®Google Scholar
• A. N. Brooks and T. J. R. Hughes, “Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations,” Comput. Meth. Appl. Mech. Eng., vol. 32, pp. 199–259, 1982.
   Web of Science ®Google Scholar
• T. J. R. Hughes, L. P. Franca, and G. M. Hulbert, “A new finite element formulation for computational fluid dynamics: VII. The Galerkin-least-squares method for advective-diffusion equations,” Comput. Meth. Appl. Mech. Eng., vol. 73, pp. 173–189, 1989.
   Web of Science ®Google Scholar
• X. H. Wu and Y. J. Dai, “Tao WQ. MLPG/SUPG method for convection-dominated problems,” Numer. Heat Transfer B, vol. 61, pp. 36–51, 2012.
   Web of Science ®Google Scholar
• T. J. R. Hughes, “Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods,” Comput. Meth. Appl. Mech. Eng., vol. 127, pp. 387–401, 1995.
   Web of Science ®Google Scholar
• T. J. R. Hughes, G. Feijoo, L. Mazzei, and J. Quincy, “The variational multiscale method-A paradigm for computational mechanics,” Comput. Meth. Appl. Mech. Eng., vol. 166, pp. 3–24, 1998. DOI: 10.1016/s0045-7825(98)00079-6.
   Web of Science ®Google Scholar
• V. John and S. Kaya, “A finite element variational multiscale method for the Navier–Stokes equations, SIAM,” J. Sci. Comput., vol. 26, pp. 1485–1503, 2005.
   Web of Science ®Google Scholar
• V. John and S. Kaya, “Finite element error analysis of a variational multiscale method for the Navier–Stokes equations,” Adv. Comput. Math., vol. 28, pp. 43–61, 2008.
   Web of Science ®Google Scholar
• V. John and A. Kindl, “Variants of projection-based finite element variational multiscale methods for the simulation of turbulent flows,” Int. J. Numer. Methods Fluids, vol. 56, pp. 1321–1328, 2008. DOI: 10.1002/fld.1712.
   Web of Science ®Google Scholar
• V. John, S. Kaya, and A. Kindl, “Finite element error analysis for a projection-based variational multiscale method with nonlinear eddy viscosity,” J. Math. Anal. Appl., vol. 344, pp. 627–641, 2008. DOI: 10.1016/j.jmaa.2008.03.015.
   Web of Science ®Google Scholar
• V. John, S. Kaya, and W. Layton, “A two-level variational multiscale method for convection-dominated convection–diffusion equations,” Comput. Meth. Appl. Mech. Eng., vol. 195, pp. 4594–4603, 2006. DOI: 10.1016/j.cma.2005.10.006.
   Web of Science ®Google Scholar
• H. B. Zheng, Y. R. Hou, F. Shi, and L. N. Song, “A finite element variational multiscale method for incompressible flows based on two local Gauss integration,” J. Comput. Phys., vol. 228, pp. 5961–5971, 2009.
   Web of Science ®Google Scholar
• T. Zhang and X. L. Li, “A variational multiscale interpolating element-free Galerkin method for convection–diffusion and Stokes problems,” Eng. Anal. Bound. Elem., vol. 82, pp. 185–193, 2017. DOI: 10.1016/j.enganabound.2017.06.013.
   Web of Science ®Google Scholar
• D. Mehdi and A. Mostafa, “Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier–Stokes equation,” Comput. Meth. Appl. Mech. Eng., vol. 311, pp. 856–888, 2016. DOI: 10.1016/j.cma.2016.09.008.
   Web of Science ®Google Scholar
• S. N. Atluri, The Meshless Method (MLPG) for Domain & BIE Discretizations. Encino, CA: Technology and Science Press, 2004.
   Google Scholar
• Z. J. Chen, Z. Y. Li, W. L. Xie, and X. H. Wu, “A two-level variational multiscale meshless local Petrov–Galerkin (VMS-MLPG) method for convection–diffusion problems with large Peclet number,” Comput. Fluids, vol. 73, pp. 1–10, 2017.
   Google Scholar
• T. J. R. Hughes, M. Mallet, and A. Mizukami, “A new finite element formulation for computational fluid dynamics II beyond SUPG,: Comput. Meth. Appl. Mech. Eng., vol. 54, pp. 341–355, 1985. DOI: 10.1016/0045-7825(86)90110-6.
   Web of Science ®Google Scholar
• B. P. Leonard, “Simple high-accuracy resolution program for convection modelling of discontinuities,” Int. J. Numer. Methods Fluids, vol. 8, pp. 1291–1318, 1988. DOI: pii:S1201-9712(17)30325-9.10.1016/j.ijid.2017.12.013.
   Web of Science ®Google Scholar