Anisotropic mesh adaptation for steady convection-dominated problems based on bubble-type local mesh generation

References

• F. Alauzet, Size gradation control of anisotropic meshes, Ann. Univ. Oradea Econ. Sci. 46 (2010), pp. 181–202.
   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. Methods Appl. Mech. Eng. 32 (1982), pp. 199–259. doi: 10.1016/0045-7825(82)90071-8
   Web of Science ®Google Scholar
• A. Cangiani and E. Su¨li, The residual-free-bubble finite element method on anisotropic partitions, SIAM J. Numer. Anal. 45 (2007), pp. 1654–1678. doi: 10.1137/060658011
   Web of Science ®Google Scholar
• W. Chen, Y. Nie, W. Zhang and L. Wang, A fast local mesh generation method about high-quality node set, Jisuan Lixue Xuebao/Chin. J. Comput. Mech. 29 (2012), pp. 704–709.
   Google Scholar
• C. Dobrzynski, Adaptation de maillage anisotrope 3d et application a` l'ae´rothermique des be˘timents, Ph.D. diss., Universite˘ Pierre et Marie Curie, Paris VI, France, 2005.
   Google Scholar
• Q. Du and D. Wang, Anisotropic centroidal Voronoi tessellations and their applications, SIAM J. Sci. Comput. 26 (2005), pp. 737–761. doi: 10.1137/S1064827503428527
   Web of Science ®Google Scholar
• P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O'Riordan and G.I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman and Hall/CRC, Boca Ration, FL, 2000.
   Google Scholar
• L. Formaggia, S. Micheletti and S. Perotto, Anisotropic mesh adaptation in computational fluid dynamics: Application to the advection–diffusion–reaction and the Stokes problems, Appl. Numer. Math. 51 (2004), pp. 511–533. doi: 10.1016/j.apnum.2004.06.007
   Web of Science ®Google Scholar
• L. Formaggia and S. Perotto, New anisotropic a priori error estimates, Numer. Math. 89 (2001), pp. 641–667. doi: 10.1007/s002110100273
   Web of Science ®Google Scholar
• L. Formaggia and S. Perotto, Anisotropic error estimates for elliptic problems, Numer. Math. 94 (2003), pp. 67–92. doi: 10.1007/s00211-002-0415-z
   Web of Science ®Google Scholar
• L.P. Franca, F. Brezzi and A. Russo, Further considerations on residual-free bubbles for advective–diffusive equations, Comput. Methods Appl. Mech. Eng. 166 (1998), pp. 25–33. doi: 10.1016/S0045-7825(98)00080-2
   Web of Science ®Google Scholar
• E. Hachem, G. Jannoun, J. Veysset and T. Coupez, On the stabilized finite element method for steady convection-dominated problems with anisotropic mesh adaptation, Appl. Math. Comput. 232 (2014), pp. 581–594.
   Web of Science ®Google Scholar
• F. Hecht, Bamg: Bidimensional anisotropic mesh generator, Inria Report, 1998.
   Google Scholar
• J.C. Heinrich, P.S. Huyakorn, O.C. Zienkiewicz and A.R. Mitchell, An ‘upwind’ finite element scheme for two-dimensional convective transport equation, Int. J. Numer. Methods Eng. 11 (1977), pp. 131–143. doi: 10.1002/nme.1620110113
   Web of Science ®Google Scholar
• W.Z. Huang, Metric tensors for anisotropic mesh generation, J. Comput. Phys. 204 (2005), pp. 633–665. doi: 10.1016/j.jcp.2004.10.024
   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: VIII. The Galerkin/least-squares method for advective–diffusive equations, Comput. Methods Appl. Mech. Eng. 73 (1989), pp. 173–189. doi: 10.1016/0045-7825(89)90111-4
   Web of Science ®Google Scholar
• V. John and P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – a review, Comput. Methods Appl. Mech. Eng. 196 (2007), pp. 2197–2215. doi: 10.1016/j.cma.2006.11.013
   Web of Science ®Google Scholar
• A. Kaya, Finite difference approximations of multidimensional unsteady convection–diffusion–reaction equations, J. Comput. Phys. 285 (2015), pp. 331–349. doi: 10.1016/j.jcp.2015.01.024
   Web of Science ®Google Scholar
• A. Kaya and A. Sendur, Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid, J. Comput. Phys. 300 (2015), pp. 574–591. doi: 10.1016/j.jcp.2015.08.007
   Web of Science ®Google Scholar
• P. Laug and H. Borouchaki, The bl2d mesh generator: Beginner's guide, user's and programmer's manual, Tech. Rep. RT-0275 3, Institut National de Recherche en Informatique et Automatique (INRIA), Rocquencourt, 1996.
   Google Scholar
• Y. Liu, Y. Nie, W. Zhang and L. Wang, Node placement method by bubble simulation and its application, Comput. Model. Eng. Sci. 55 (2010), pp. 89–109.
   Web of Science ®Google Scholar
• S. Micheletti, S. Perotto and M. Picasso, Stabilized finite elements on anisotropic meshes: A priori error estimates for the advection–diffusion and the Stokes problems, SIAM J. Numer. Anal. 41 (2003), pp. 1131–1162. doi: 10.1137/S0036142902403759
   Web of Science ®Google Scholar
• H. Nguyen, M. Gunzburger, L. Ju and J. Burkardt, Adaptive anisotropic meshing for steady convection-dominated problems, Comput. Methods. Appl. Mech. Eng. 198 (2009), pp. 2964–2981. doi: 10.1016/j.cma.2009.05.001
   Web of Science ®Google Scholar
• M. Picasso, Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz–Zhu error estimator, Commun. Numer. Methods Eng. 19 (2003), pp. 13–23. doi: 10.1002/cnm.546
   Google Scholar
• H.G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Vol. 24, Springer Science & Business Media, Berlin, 2008.
   Google Scholar
• R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comput. 77 (2008), pp. 227–241. doi: 10.1090/S0025-5718-07-01959-X
   Web of Science ®Google Scholar
• P. Sun, L. Chen and J. Xu, Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems, J. Sci. Comput. 43 (2010), pp. 24–43. doi: 10.1007/s10915-009-9337-6
   Web of Science ®Google Scholar
• D. Wang, R. Li and N. Yan, An edge-based anisotropic mesh refinement algorithm and its application to interface problems, Commun. Comput. Phys. 8 (2010), pp. 511–540.
   Web of Science ®Google Scholar
• H. Xie and X. Yin, Metric tensors for the interpolation error and its gradient in lp norm, J. Comput. Phys. 256 (2014), pp. 543–562. doi: 10.1016/j.jcp.2013.09.008
   Google Scholar
• Z. Zhang, Finite element superconvergence on Shishkin mesh for 2-d convection–diffusion problems, Math. Comput. 72 (2003), pp. 1147–1177. doi: 10.1090/S0025-5718-03-01486-8
   Web of Science ®Google Scholar
• W. Zhang, Y. Nie and Y. Gu, Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation, J. Comput. Appl. Math. 280 (2015), pp. 42–58. doi: 10.1016/j.cam.2014.11.049
   Web of Science ®Google Scholar
• W. Zhang, Y. Nie and Y. Li, A new anisotropic local meshing method and its application in parametric surface triangulation, Comput. Model. Eng. Ences. 88 (2012), pp. 507–529.
   Web of Science ®Google Scholar
• O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Methods Eng. 33 (1992), pp. 1331–1364. doi: 10.1002/nme.1620330702
   Web of Science ®Google Scholar
• O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J. Numer. Methods Eng. 33 (2010), pp. 1365–1382. doi: 10.1002/nme.1620330703
   Google Scholar