A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions Part I: Second order linear PDE
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1193-1222.

We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving second order linear PDEs. Our residual type upper and lower bound error estimates measure the error in the energy norm. The main merit of our method is that the method is parameter-free, in the sense that all but one solution-dependent constants appearing in the upper and lower bound estimates are explicitly computable by solving local eigenvalue problems, and the only non-computable constant can be reasonably approximated by a computable one without affecting the overall effectiveness of the estimates in practice. As a side product of our formulation, the penalty parameter in the interior penalty formulation can be automatically determined as well. We develop an efficient numerical procedure to compute the error estimators. Numerical results for a variety of problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective.

DOI : 10.1051/m2an/2015069
Classification : 65J10, 65N15, 65N30
Mots clés : Discontinuous Galerkin method, a posteriori error estimation, non-polynomial basis functions, partial differential equations
Lin, Lin 1 ; Stamm, Benjamin 2

1 Department of Mathematics, University of California Berkeley and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.
2 Sorbonne Universités, UPMC Univ. Paris 06, UMR 7598, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
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Lin, Lin; Stamm, Benjamin. A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions Part I: Second order linear PDE. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1193-1222. doi : 10.1051/m2an/2015069. http://www.numdam.org/articles/10.1051/m2an/2015069/

M. Ainsworth and R. Rankin, Technical note: A note on the selection of the penalty parameter for discontinuous Galerkin finite element schemes. Numer. Methods Partial Differ. Eq. 28 (2012) 1099–1104. | DOI | MR

M. Amara, R. Djellouli and C. Farhat, Convergence analysis of a discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of Helmholtz problems. SIAM J. Numer. Anal. 47 (2009) 1038–1066. | DOI | MR | Zbl

D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. | DOI | MR | Zbl

I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863–875. | DOI | MR | Zbl

G.A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31 (1977) 45–59. | DOI | MR | Zbl

C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311–341. | DOI | MR | Zbl

J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin Methods. Computing Methods in Applied Sciences (Second Internat. Sympos., Versailles, 1975). In vol. 58 of Lect. Notes Phys. (1976) 207–216. | MR

Y. Epshteyn and B. Rivière, Estimation of penalty parameters for symmetric interior penalty Galerkin methods. J. Comput. Appl. Math. 206 (2007) 843–872. | DOI | MR | Zbl

M.J. Frisch, J.A. Pople and J.S. Binkley, Self-consistent molecular orbital methods 25. Supplementary functions for Gaussian basis sets. J. Chem. Phys. 80 (1984) 3265–3269. | DOI

S. Giani and E.J.C. Hall, An a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems. Math. Models Methods Appl. Sci. 22 (2012) 1250030–1250064. | DOI | MR | Zbl

P. Henning and M. Ohlberge, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems. Discrete Contin. Dyn. Systems Series S 8 (2015) 119–150. | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49 (2011) 264–284. | DOI | MR | Zbl

T.Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. | DOI | MR | Zbl

P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002) 2133–2163. | DOI | MR | Zbl

P. Houston, D. Schötzau and T.P. Wihler, Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. Models Methods Appl. Sci. 17 (2007) 33–62. | DOI | MR | Zbl

J. Junquera, O. Paz, D. Sanchez-Portal and E. Artacho, Numerical atomic orbitals for linear-scaling calculations. Phys. Rev. B 64 (2001) 235111–235119. | DOI

O.A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399. | DOI | MR | Zbl

J. Kaye, L. Lin and C. Yang, A posteriori error estimator for adaptive local basis functions to solve Kohn–Sham density functional theory. Commun. Math. Sci. 13 (2015) 1741–1773. | DOI | MR | Zbl

A.V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comp. 23 (2001) 517. | DOI | MR | Zbl

L. Lin, J. Lu, L. Ying and E. Weinan, Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation. J. Comput. Phys. 231 (2012) 2140–2154. | DOI | Zbl

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. | DOI | MR | Zbl

M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems. Multiscale Model. Simul. 4 (2005) 88–114. | DOI | MR | Zbl

M. Ohlberger and F. Schindler, A-posteriori error estimates for the localized reduced basis multi-scale method. In Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Edited by J. Fuhrmann, M. Ohlberger and Ch. Rohde. Vol. 77 of Springer Proc. Math. Stat. Springer International Publishing (2014) 421–429. | MR | Zbl

D. Schötzau and L. Zhu, A robust a posteriori error estimator for discontinuous Galerkin methods for convection–diffusion equations. Appl. Numer. Math. 59 (2009) 2236–2255. | DOI | MR | Zbl

C. Schwab, p-and hp-Finite Element Methods. Oxford University Press, New York (1998). | MR | Zbl

B. Stamm and T. Wihler, hp-Optimal discontinuous Galerkin methods for linear elliptic problems. Math. Comput. 79 (2010) 2117–2133. | DOI | MR | Zbl

R. Tezaur and C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Meth. Eng. 66 (2006) 796–815. | DOI | MR | Zbl

E. Weinan and B. Engquist, The heterognous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. | DOI | MR | Zbl

M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. | DOI | MR | Zbl

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