Abstract
We consider a standard Adaptive weak Galerkin (AWG) finite element method for second order elliptic problems. We prove that the sum of the energy error and the scaled error estimator of AWG method, between two consecutive adaptive loops, is a contraction. At last, we present some numerical experiments to support the theoretical results.
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Binev, P., Dahmen, W., DeVore, R.A.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)
Cascon, 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)
Chen, L.: iFEM: An Integrated Finite Element Methods Package in MATLAB. University of California, Irvine (2009)
Chen, L., Holst, M., Xu, J.: Convergence and optimality of adaptive mixed finite element methods. Math. Comput. 78(265), 35–53 (2009)
Chen, L., Wang, J., Ye, X.: A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput. 59(2), 496–511 (2014)
Du, Y., Zhang, Z.: A numerical analysis of the weak Galerkin method for the Helmholtz equation with high wave number. Commun. Comput. Phys. 22(1), 133–156 (2017)
Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)
Huang, J., Xu, Y.: Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation. Sci. China Math. 55(5), 1083–1098 (2012)
Kellogg, R.B.: On the Poisson equation with intersecting interfaces. Appl. Anal. 4(2), 101–129 (1974)
Li, H.G., Mu, L., Ye, X.: A posteriori error estimates for the weak Galerkin finite element methods on polytopal meshes. Commun. Comput. Phys. 26(2), 558–578 (2019)
Lin, G., Liu, J., Mu, L., Ye, X.: Weak Galerkin finite element methods for Darcy flow: anisotropy and heterogeneity. J. Comput. Phys. 276, 422–437 (2014)
Mitchell, W.F.: A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Softw. 15(4), 326–347 (1989)
Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2000)
Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (2002)
Mu, L.: Weak Galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes. J. Comput. Appl. Math. 361, 413–425 (2019)
Mu, L., Wang, J., Wei, G., Zhao, S.: Weak Galerkin methods for second order elliptic interface problems. J. Comput. Phys. 250, 106–125 (2013)
Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer. Methods Part. Differ. Equ. 30(3), 1003–1029 (2014)
Mu, L., Wang, J., Ye, X.: A new weak Galerkin finite element method for the Helmholtz equation. IMA J. Numer. Anal. 35(3), 1228–1255 (2015)
Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element method with polynomial reduction. J. Comp. Appl. Math. 285, 45–58 (2015)
Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on polytopal meshes. Int. J. Numer. Anal. Mod. 12(1), 31–53 (2015)
Mu, L., Wang, J., Ye, X., Zhang, S.: A \(C^{0}\)-weak Galerkin finite element method for the biharmonic equation. J. Sci. Comput. 59(2), 473–495 (2014)
Mu, L., Wang, J., Ye, X., Zhao, S.: A numerical study on the weak Galerkin method for the Helmholtz equation. Commun. Comput. Phys. 15(5), 1461–1479 (2014)
Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Devore, R.A., Kunoth, A. (eds.) Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)
Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comp. 77(261), 227–241 (2008)
Wang, C., Wang, J.: An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. Comput. Math. Appl. 68(12), 2314–2330 (2013)
Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241(0), 103–115 (2013)
Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83(289), 2101–2126 (2014)
Zhang, T., Chen, Y.: A posteriori error analysis for the weak Galerkin method for solving elliptic problems. Int. J. Comput. Methods 15(8), 1850075 (2018)
Zhang, J., Zhang, K., Li, J., Wang, X.: A weak Galerkin finite element method for the Navier–Stokes equations. Commun. Comput. Phys. 23, 706–746 (2018)
Zhang, T., Lin, T.: A posteriori error estimate for a modified weak Galerkin method solving elliptic problems. Numer. Methods Part. D. E 33(1), 381–398 (2017)
Zheng, X., Xie, X.: A posteriori error estimator for a weak Galerkin finite element solution of the Stokes problem. E. Asian. J. Appl. Math. 7(3), 508–529 (2017)
Zhong, L., Chen, L., Shu, S., Wittum, G., Xu, J.: Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations. Math. Comput. 81(278), 623–642 (2012)
Zhong, L., Shu, S., Chen, L., Xu, J.: Convergence of adaptive edge finite element methods for H(curl)-elliptic problems. Numer. Linear Algebra Appl. 17(2–3), 415–432 (2010)
Acknowledgements
The authors would like to thank Professor Long Chen, University of California at Irvine, for providing many constructive suggestions.
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This research was supported in part by supported by the National Natural Science Foundation of China (Nos. 11671159, 12071160), the Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515010724), the Characteristic Innovation Projects of Guangdong colleges and universities, China (No. 2018KTSCX044) and the General Project topic of Science and Technology in Guangzhou, China (No. 201904010117).
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Xie, Y., Zhong, L. Convergence of Adaptive Weak Galerkin Finite Element Methods for Second Order Elliptic Problems. J Sci Comput 86, 17 (2021). https://doi.org/10.1007/s10915-020-01387-7
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DOI: https://doi.org/10.1007/s10915-020-01387-7