Abstract
A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric which is often based on some type of Hessian recovery. This study discusses the use of a hierarchical basis error estimator for the development of an anisotropic metric tensor needed for the adaptive finite element solution. A global hierarchical basis error estimator is employed to obtain reliable directional information. Numerical results for a selection of different applications show that the method performs comparable with existing metric tensors based on Hessian recovery and can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Agouzal, A., Lipnikov, K., Vassilevski, Y.: Generation of quasi-optimal meshes based on a posteriori error estimates. In: Proceedings of the 16th International Meshing Roundtable, pp. 139–148 (2008)
Agouzal, A., Lipnikov, K., Vassilevski, Y.: Anisotropic mesh adaptation for solution of finite element problems using hierarchical edge-based error estimates. In: Proceedings of the 18th International Meshing Roundtable, pp. 595–610 (2009)
Agouzal, A., Lipnikov, K., Vassilevski, Y.: Hessian-free metric-based mesh adaptation via geometry of interpolation error. Comput. Math. Math. Phys. 50(1), 124–138 (2010)
Agouzal, A., Vassilevski, Y.V.: Minimization of gradient errors of piecewise linear interpolation on simplicial meshes. Comput. Methods Appl. Mech. Engrg. 199(33-36), 2195–2203 (2010)
Apel, T., Grosman, S., Jimack, P.K., Meyer, A.: A new methodology for anisotropic mesh refinement based upon error gradients. Appl. Numer. Math. 50(3-4), 329–341 (2004)
Bank, R.E., Kent Smith, R.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30(4), 921–935 (1993)
Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part I: Grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294–2312 (2003)
Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part II: General unstructured grids. SIAM J. Numer. Anal. 41(6), 2313–2332 (2003)
Bildhauer, M.: Convex variational problems. Linear, nearly linear and anisotropic growth conditions. Lecture Notes in Mathematics, vol. 1818. Springer, Heidelberg (2000)
Borouchaki, H., George, P.L., Hecht, F., Laug, P., Saltel, E.: Delaunay mesh generation governed by metric specifications. Part I. Algorithms. Finite Elements in Analysis and Design 25(1-2), 61–83 (1997)
Borouchaki, H., George, P.L., Mohammadi, B.: Delaunay mesh generation governed by metric specifications. Part II. Applications. Finite Elem. Anal. Des. 25(1-2), 85–109 (1997)
Cao, W., Huang, W., Russell, R.D.: Comparison of two-dimensional r-adaptive finite element methods using various error indicators. Math. Comput. Simulation 56(2), 127–143 (2001)
Castro-Díaz, M.J., Hecht, F., Mohammadi, B., Pironneau, O.: Anisotropic unstructured mesh adaption for flow simulations. Int. J. Numer. Meth. Fluids 25(4), 475–491 (1997)
Ciarlet, P.G., Raviart, P.A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Engrg. 2(1), 17–31 (1973)
Deuflhard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. Impact Comput. Sci. Engrg. 1(1), 3–35 (1989)
Dobrowolski, M., Gräf, S., Pflaum, C.: On a posteriori error estimators in the finite element method on anisotropic meshes. Electron. Trans. Numer. Anal. 8, 36–45 (1999)
Dolejší, V.: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci. 1(3), 165–178 (1998)
Dörfler, W., Nochetto, R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91, 1–12 (2002)
Formaggia, L., Perotto, S.: New anisotropic a priori error estimates. Numer. Math. 89(4), 641–667 (2001)
Frey, P.J., George, P.-L.: Mesh Generation, 2nd edn. John Wiley & Sons, Inc., Hoboken (2008)
Hecht, F.: BAMG: Bidimensional Anisotropic Mesh Generator (2006), Source code: http://www.ann.jussieu.fr/~hecht/ftp/bamg/
Huang, W.: Measuring mesh qualities and application to variational mesh adaptation. SIAM J. Sci. Comput. 26(5), 1643–1666 (2005)
Huang, W.: Metric tensors for anisotropic mesh generation. J. Comput. Phys. 204(2), 633–665 (2005)
Huang, W., Kamenski, L., Lang, J.: A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates. J. Comput. Phys. 229(6), 2179–2198 (2010)
Huang, W., Kamenski, L., Li, X.: Anisotropic mesh adaptation for variational problems using error estimation based on hierarchical bases. Canad. Appl. Math. Quart. (to appear, 2010) arXiv:1006.0191
Huang, W., Li, X.: An anisotropic mesh adaptation method for the finite element solution of variational problems. Finite Elem. Anal. Des. 46(1-2), 61–73 (2010)
Huang, W., Sun, W.W.: Variational mesh adaptation II: error estimates and monitor functions. J. Comput. Phys. 184(2), 619–648 (2003)
Kamenski, L.: Anisotropic Mesh Adaptation Based on Hessian Recovery and A Posteriori Error Estimates. PhD thesis, TU Darmstadt (2009)
Li, X., Huang, W.: An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems. J. Comput. Phys. (to appear, 2010) arXiv:1003.4530
Ovall, J.S.: The dangers to avoid when using gradient recovery methods for finite element error estimation and adaptivity. Technical Report 6, Max Planck Institute for Mathematics in the Sciences (2006)
Ovall, J.S.: Function, gradient, and Hessian recovery using quadratic edge-bump functions. SIAM J. Numer. Anal. 45(3), 1064–1080 (2007)
Pardo, D., Demkowicz, L.: Integration of hp-adaptivity and a two-grid solver for elliptic problems. Comput. Methods Appl. Mech. Engrg. 195(7-8), 674–710 (2006)
Zhang, Z., Naga, A.: A new finite element gradient recovery method: Superconvergence property. SIAM J. Sci. Comput. 26(4), 1192–1213 (2005)
Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Methods Engrg. 33(7), 1331–1364 (1992)
Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. Int. J. Numer. Methods Engrg. 33(7), 1365–1382 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kamenski, L. (2010). A Study on Using Hierarchical Basis Error Estimates in Anisotropic Mesh Adaptation for the Finite Element Method. In: Shontz, S. (eds) Proceedings of the 19th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15414-0_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-15414-0_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15413-3
Online ISBN: 978-3-642-15414-0
eBook Packages: EngineeringEngineering (R0)