Abstract
In this paper, a contraction property is proved for an adaptive finite element method for controlling the global L 2 error on convex polyhedral domains. Furthermore, it is shown that the method converges in L 2 with the best possible rate. The method that is analyzed is the standard adaptive method except that, if necessary, additional refinements are made to keep the meshes sufficiently mildly graded. This modification does not compromise the quasi-optimality of the resulting algorithm.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Babuška I., Osborn J.: Analysis of finite element methods for second order boundary value problems using mesh dependent norms. Numer. Math. 34, 41–62 (1980)
Binev P., Dahmen W., DeVore R.: Adaptive finite element methods with convergence rates. Numer. Math. 97, 219–268 (2004)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)
Cascon J., Kreuzer C., Nochetto R.H., Siebert K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46, 2524–2550 (2008)
Demlow A.: Convergence of an adaptive finite element method for controlling local energy errors. SIAM J. Numer. Anal. 48(2), 470–497 (2010)
Dörfler W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)
Eriksson K.: An adaptive finite element method with efficient maximum norm error control for elliptic problems. Math. Models Methods Appl. Sci. 4, 313–329 (1994)
Eriksson K., Johnson C.: Adaptive finite element methods for parabolic problems. II. Optimal error estimates in L ∞ L 2 and L ∞ L ∞. SIAM J. Numer. Anal. 32, 706–740 (1995)
Liao X., Nochetto R.H.: Local a posteriori error estimates and adaptive control of pollution effects. Numer. Methods Partial Differ. Equ. 19, 421–442 (2003)
Maubach J.: Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Comput. 16, 210–227 (1995)
Mekchay K., Nochetto R.H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43, 1803–1827 (2005) (electronic)
Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44, 631–658 (2002) (electronic) (2003) Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38(2), 466–488 (2000) (electronic); MR1770058 (2001g:65157)]
Morin, P., Siebert, K., Veeser, A.: Convergence of Finite Elements Adapted for Weaker Norms. Applied and Industrial Mathematics in Italy II. Ser. Adv. Math. Appl. Sci., vol. 75, pp. 468–479. World Sci. Publ., Hackensack, NJ (2007)
Morin P., Siebert K., Veeser A.: A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18, 707–737 (2008)
Nitsche J.A., Schatz A.H.: Interior estimates for Ritz-Galerkin methods. Math. Comput. 28, 937–958 (1974)
Nochetto, R., Siebert, K., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: DeVore, R., Kunoth, A. (eds.) Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday, pp. 409–542. Springer, Berlin (2009)
Schmidt, A., Siebert, K.G.: Design of adaptive finite element software. The finite element toolbox ALBERTA, with 1 CD-ROM (Unix/Linux). Lecture Notes in Computational Science and Engineering, vol. 42, pp. xii+315. Springer, Berlin (2005)
Scott L.R., Zhang S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)
Stevenson R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7, 245–269 (2007)
Stevenson R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77, 227–241 (2008)
Traxler C.T.: An algorithm for adaptive mesh refinement in n dimensions. Computing 59, 115–137 (1997)
Verfürth R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester (1996)
Wihler T.P.: Weighted L 2-norm a posteriori error estimation of FEM in polygons. Int. J. Numer. Anal. Model. 4, 100–115 (2007)
Xu J., Zhou A.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69, 881–909 (2000)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
A. Demlow was partially supported by National Science Foundation Grant DMS-0713770.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Demlow, A., Stevenson, R. Convergence and quasi-optimality of an adaptive finite element method for controlling L 2 errors. Numer. Math. 117, 185–218 (2011). https://doi.org/10.1007/s00211-010-0349-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-010-0349-9