Least-squares RBF-FD method for the incompressible Stokes equations with the singular source

References

• C. Bernardi, et al., “A model for two coupled turbulent fluids part I: Analysis of the system,” Jacques-Louis Lions, vol. 31, no. 2, pp. 69–102, 2002.
   Google Scholar
• C. Bernardi, et al., “A model for two coupled turbulent fluids part II: Numerical analysis of a spectral discretization,” SIAM Journal on Numerical Analysis, vol. 40, no. 6, pp. 2368–2394, 2002. DOI: 10.1137/S0036142901385829.
   Web of Science ®Google Scholar
• C. Bernardi, et al., “A model for two coupled turbulent fluids part III: Numerical approximation by finite elements,” Numerische Mathematik, vol. 98, no. 1, pp. 33–66, 2004. DOI: 10.1007/s00211-003-0490-9.
   Web of Science ®Google Scholar
• J. Kim, D. Kim, and H. Choi, “An immersed-boundary finite-volume method for simulations of flow in complex geometries,” Journal of Computational Physics, vol. 171, no. 1, pp. 132–150, 2001. DOI: 10.1006/jcph.2001.6778.
   Web of Science ®Google Scholar
• L. Lee, and R. J. LeVeque, “An immersed interface method for incompressible Navier–Stokes equations,” SIAM Journal on Scientific Computing, vol. 25, no. 3, pp. 832–856, 2003. DOI: 10.1137/S1064827502414060.
   Web of Science ®Google Scholar
• R. J. LeVeque, and Z. Li, “Immersed interface methods for Stokes flow with elastic boundaries or surface tension,” SIAM Journal on Scientific Computing, vol. 18, no. 3, pp. 709–735, 1997. DOI: 10.1137/S1064827595282532.
   Web of Science ®Google Scholar
• Z. Li, and M. C. Lai, “The immersed interface method for the Navier-Stokes equations with singular forces,” Journal of Computational Physics, vol. 171, no. 2, pp. 822–842, 2001. DOI: 10.1006/jcph.2001.6813.
   Web of Science ®Google Scholar
• A. Pinelli, et al., “Immersed-boundary methods for general finite-difference and finite-volume Navier–Stokes solvers,” Journal of Computational Physics, vol. 229, no. 24, pp. 9073–9091, 2010. DOI: 10.1016/j.jcp.2010.08.021.
   Web of Science ®Google Scholar
• R. E. Ewing. The Mathematics of Reservoir Simulation. Philadelphia: SIAM, 1983.
   Google Scholar
• G. Zhang, H. Su, and X. Feng, “A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem,” Numerical Heat Transfer, Part B: Fundamentals, vol. 73, no. 5, pp. 329–341, 2018. DOI: 10.1080/10407790.2018.1486647.
   Web of Science ®Google Scholar
• S. Zhao, et al., “Two types of spurious oscillations at layers diminishing methods for convection–diffusion–reaction equations on surface,” Numerical Heat Transfer, Part A: Applications, vol. 74, no. 7, pp. 1387–1404, 2018. DOI: 10.1080/10407782.2018.1538292.
   Web of Science ®Google Scholar
• C. S. Peskin, “Numerical analysis of blood flow in the heart,” Journal of Computational Physics, vol. 25, no. 3, pp. 220–252, 1977. DOI: 10.1016/0021-9991(77)90100-0.
   Web of Science ®Google Scholar
• C. S. Peskin, “The immersed boundary method,” Acta Numerica, vol. 11, pp. 479–517, 2002.
   Google Scholar
• R. Teman, Navier–Stokes Equations: Theory and Numerical Analysis. New York: North-Holland, 1977.
   Google Scholar
• J. Li, et al., “Two-level meshless local Petrov Galerkin method for multi-dimensional nonlinear convection–diffusion equation based on radial basis function,” Numerical Heat Transfer, Part B: Fundamentals, vol. 74, no. 4, pp. 1–14, 2019.
   Web of Science ®Google Scholar
• T. Lin, et al., “A rectangular immersed finite element space for interface problems,” Scientific Computing & Applications, vol. 7, pp. 107–114, 2001.
   Google Scholar
• V. Bayona, et al., “RBF-FD formulas and convergence properties,” Journal of Computational Physics, vol. 229, no. 22, pp. 8281–8295, 2010. DOI: 10.1016/j.jcp.2010.07.008.
   Web of Science ®Google Scholar
• B. Vaughan, B. Smith, and D. Chopp, “A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources,” Communications in Applied Mathematics and Computational Science, vol. 1, no. 1, pp. 207–228, 2006. DOI: 10.2140/camcos.2006.1.207.
   Google Scholar
• W. Wang, J. Wu, and X. Feng, “A novel pressure-correction projection finite element method for incompressible natural convection problem with variable density,” Numerical Heat Transfer, Part A: Applications, vol. 74, no. 2, pp. 1001–1017, 2018. DOI: 10.1080/10407782.2018.1505093.
   Web of Science ®Google Scholar
• Y. Wang, H. Su, and X. Feng, “Streamline diffusion finite element method for stationary incompressible natural convection problem,” Numerical Heat Transfer, Part B: Fundamentals, vol. 74, no. 2, pp. 519–537, 2018. DOI: 10.1080/10407790.2018.1513281.
   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,” Numerical Heat Transfer, Part A: Applications, vol. 68, no. 7, pp. 777–796, 2015. DOI: 10.1080/10407782.2015.1012851.
   Web of Science ®Google Scholar
• Y. C. Zhou, et al., “High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources,” Journal of Computational Physics, vol. 213, no. 1, pp. 1–30, 2006. DOI: 10.1016/j.jcp.2005.07.022.
   Web of Science ®Google Scholar
• G. E. Fasshauer, Meshfree Approximation Methods with MATLAB. World Scientific, 2007.
   Google Scholar
• R. J. LeVeque, and Z. Li, “The immersed interface method for elliptic equations with discontinuous coefficients and singular sources,” SIAM Journal on Numerical Analysis, vol. 31, no. 4, pp. 1019–1044, 1994. DOI: 10.1137/0731054.
   Web of Science ®Google Scholar
• V. Bayona, M. Moscoso, and M. Kindelan, “Optimal constant shape parameter for multiquadric based RBF-FD method,” Journal of Computational Physics, vol. 230, no. 19, pp. 7384–7399, 2011. DOI: 10.1016/j.jcp.2011.06.005.
   Web of Science ®Google Scholar
• A. Quarteroni, and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford: Oxford University Press, 1999.
   Google Scholar
• N. Li, et al., “Multiquadric RBF-FD method for the convection-dominated diffusion problems base on Shishkin nodes,” International Journal of Heat and Mass Transfer, vol. 118, pp. 734–745, 2018. DOI: 10.1016/j.ijheatmasstransfer.2017.11.011.
   Web of Science ®Google Scholar
• Y. Qiao, J. Zhao, and X. Feng, “A compact integrated RBF method for time fractional convection-diffusion-reaction equations,” Computers & Mathematics with Applications, vol. 77, no. 9, pp. 2263–2278, 2019. DOI: 10.1016/j.camwa.2018.12.017.
   Web of Science ®Google Scholar
• P. Hessari, “First order system least squares method for the interface problem of the Stokes equations,” Computers & Mathematics with Applications, vol. 68, no. 3, pp. 309–324, 2014. DOI: 10.1016/j.camwa.2014.06.003.
   Web of Science ®Google Scholar
• P. Hessari, B. C. Shin, and B. Jang, “Analysis of least squares pseudo-spectral method for the interface problem of the Navier–Stokes equations,” Computers & Mathematics with Applications, vol. 69, no. 8, pp. 838–851, 2015. DOI: 10.1016/j.camwa.2015.01.015.
   Web of Science ®Google Scholar
• S-u-I. Masood A, “Meshless analysis of parabolic interface problems,” Engineering Analysis with Boundary Elements, vol. 94, pp. 134–152, 2018.
   Web of Science ®Google Scholar
• B. C. Shin, and J. H. Jung, “Spectral collocation and radial basis function methods for one-dimensional interface problems,” Applied Numerical Mathematics, vol. 61, no. 8, pp. 911–928, 2011. DOI: 10.1016/j.apnum.2011.03.005.
   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,” Numerical Heat Transfer, Part B: Fundamentals, vol. 72, no. 5, pp. 349–360, 2017. DOI: 10.1080/10407790.2017.1409511.
   Web of Science ®Google Scholar
• H. Wendland, Scattered Data Approximation. Cambridge: Cambridge University Press, 2004.
   Google Scholar
• J. Li, et al., “Meshless local Petrov Galerkin method for 2D/3D nonlinear convection-diffusion equations based on LS-RBF-PUM,” Numerical Heat Transfer, Part B: Fundamentals, vol. 74, no. 1, pp. 450–464, 2018. DOI: 10.1080/10407790.2018.1515331.
   Web of Science ®Google Scholar
• J. Li, X. Feng, and Y. He, “RBF-based meshless local Petrov Galerkin method for the multi-dimensional convection–diffusion–reaction equation,” Engineering Analysis with Boundary Elements, vol. 98, pp. 46–53, 2019. DOI: 10.1016/j.enganabound.2018.10.003.
   Web of Science ®Google Scholar
• Y. Qiao, S. Zhai, and X. Feng, “RBF-FD method for the high dimensional time fractional convection-diffusion equation,” International Communications in Heat and Mass Transfer, vol. 89, pp. 230–240, 2017. DOI: 10.1016/j.icheatmasstransfer.2017.08.016.
   Web of Science ®Google Scholar