Abstract
We analyze an adaptive discontinuous finite element method (ADFEM) for the weakly over-penalized symmetric interior penalty (WOPSIP) operator applied to symmetric positive definite second order elliptic boundary value problems. For first degree polynomials, we prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive loops of the adaptive algorithm. After establishing this geometric decay, we define a suitable approximation class and prove that the adaptive WOPSIP method obeys a quasi-optimal rate of convergence.
Similar content being viewed by others
References
Ainsworth, M., Oden, J.T.:. A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics. Wiley-Interscience, Wiley, New York, NY (2000)
Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin methods for elliptic problems. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods, pp. 89–101. Springer, Berlin (2000)
Arnold, D.N., Brezzi, F. , Cockburn, B. , Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001/2002)
Barker, A.T., Brenner, S.C., Park, E.-H., Sung, L.-Y.: Two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method. J. Sci. Comput. 47, 27–49 (2011)
Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)
Bonito, A., Nochetto, R.H.: Quasi-optimal convergence rate of an adaptive discontinuous galerkin method. SIAM J. Numer. Anal. 48(2), 734–771 (2010)
Brenner, S.C.: Poincarè-Friedrichs inequalities for piecewise \({H}^1\) functions. SIAM J. Numer. Anal. 41, 306–324 (2003)
Brenner, S.C., Gudi, T., Owens, L., Sung, L.-Y.: An intrinsically parallel finite element method. J. Sci. Comput. 42(1), 118–121 (2010)
Brenner, S.C., Gudi, T., Sung, L.-Y.: A posteriori error control for a weakly over-penalized symmetric interior penalty method. J. Sci. Comput. 666, 666–666 (2009)
Brenner, S.C., Owens, L., Sung, L.-Y.: A weakly over-penalized symmetric interior penalty method. Electron. Trans. Numer. Anal. 30, 107–127 (2008)
Brenner, S.C., Owens, L., Sung, L.-Y.: Higher order weakly over-penalized symmetric interior penalty methods. J. Comput. Appl. Math. 236, 2883–2894 (2012)
Brenner, S.C., Park, E.-H., Sung, L.-Y.: A bddc preconditioner for a weakly over-penalized symmetric interior penalty method. Numer. Linear Alg. Appl. (2012). doi:10.1002/nla.1838
Cascón, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)
Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)
Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003). (electronic)
Maubach, J.M.: Local bisection refinement for \(n\)-simplicial grids generated by reflection. SIAM J. Sci. Comput. 16(1), 210–227 (1995)
Morin, P., Nochetto, R.H. , Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (electronic) (2003), (2002)
Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)
Owens, L.: Multigrid Methods for Weakly Over-Penalized Interior Penalty Methods. PhD thesis, University of South Carolina (2007)
Shewchuk, J.R.: Triangle: engineering a 2D quality mesh generator and Delaunay Triangulator. In: Lin, M.C., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering, vol. 1148, pp. 203–222. Lecture Notes in Computer ScienceFrom the First ACM Workshop on Applied Computational Geometry, Springer, Berlin (1996)
Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77(261), 227–241 (2008). (electronic)
Stevenson, R.P.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007)
Wheeler, M.F.: An elliptic collocation-finite-element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work of this author was partially supported by Dod-Navy-Office Of Naval Research under Award Number N000140811113.
Rights and permissions
About this article
Cite this article
Owens, L. Quasi-Optimal Convergence Rate of an Adaptive Weakly Over-Penalized Interior Penalty Method. J Sci Comput 59, 309–333 (2014). https://doi.org/10.1007/s10915-013-9765-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-013-9765-1