A gradient recovery method based on an oblique projection and boundary modification

Muhammad Ilyas, Bishnu P. Lamichhane, Michael H. Meylan


The gradient recovery method is a technique to improve the approximation of the gradient of a solution by using post-processing methods. We use an $L^2$-projection based on an oblique projection, where the trial and test spaces differ, for efficient numerical computation. We modify our oblique projection by applying the boundary modification method to obtain higher order approximation on the boundary patch. Numerical examples are presented to demonstrate the efficiency and optimality of the approach.

  • H. Guo, Z. Zhang, R. Zhao and Q. Zou. Polynomial preserving recovery on boundary. J. Comput. Appl. Math., 307:119–133, 2016. doi:10.1016/j.cam.2016.03.003
  • M. Ilyas and B. P. Lamichhane. A stabilised mixed finite element method for the Poisson problem based on a three-field formulation. In editors M. Nelson, D. Mallet, B. Pincombe and J. Bunder Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C177–C192, September 2016. doi:10.21914/anziamj.v57i0.10356
  • M. Krizek and P. Neittaanmaki. Superconvergence phenomenon in the finite element method arising from averaging gradients. Numerische Mathematik, 45(1):105–116, 1984. doi:10.1007/BF01379664
  • B. P. Lamichhane. A gradient recovery operator based on an oblique projection. Electron. Trans. Numer. Anal., 37:166–172, 2010. http://emis.ams.org/journals/ETNA/vol.37.2010/pp166-172.dir/pp166-172.pdf
  • B. P. Lamichhane. Mixed finite element methods for the Poisson equation using biorthogonal and quasi-biorthogonal systems. Advances in Numerical Analysis, 2013:189045, 2013. doi:10.1155/2013/189045
  • B. P. Lamichhane. A finite element method for a biharmonic equation based on gradient recovery operators. BIT Numerical Mathematics, 54(2):469–484, 2014. doi:10.1007/s10543-013-0462-0
  • B. P. Lamichhane, R. P. Stevenson and B. I. Wohlmuth. Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numerische Mathematik, 102(1):93–121, 2005. doi:10.1007/s00211-005-0636-z
  • J. Xu and Z. Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comp., 73(247):1139–1152, 2004. doi:10.1090/S0025-5718-03-01600-4
  • Z. Zhang and A. Naga. A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput., 26(4):1192–1213, 2005. doi:10.1137/S1064827503402837
  • O. C. Zienkiewicz and J. Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique. Internat. J. Numer. Methods Engrg., 33(7):1331–1364, 1992. doi:10.1002/nme.1620330702


Gradient recovery, oblique projection, boundary modification

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DOI: http://dx.doi.org/10.21914/anziamj.v58i0.11730

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