New superconvergent structures developed from the finite volume element method in 1D

Wang, Xiang; Lv, Junliang; Li, Yonghai

doi:10.1090/mcom/3587

New superconvergent structures developed from the finite volume element method in 1D
HTML articles powered by AMS MathViewer

by Xiang Wang, Junliang Lv and Yonghai Li;

Math. Comp. 90 (2021), 1179-1205

DOI: https://doi.org/10.1090/mcom/3587

Published electronically: December 9, 2020

HTML | PDF | Request permission

Abstract:

New superconvergence structures are introduced by the finite volume element method (FVEM), which gives us the freedom to choose the superconvergent points of the derivative (for odd order schemes) and the superconvergent points of the function value (for even order schemes) for $k\geq 3$. The general orthogonal condition and the modified M-decomposition (MMD) technique are established to prove the superconvergence properties of the new structures. In addition, the relationships between the orthogonal condition and the convergence properties for the FVE schemes are carried out in Table 1. The above results are also valid to $k=1,2$. For these cases, the new superconvergence structures are the same with the Gauss-Lobatto structure, which yields to the FVE schemes with Gauss-points-dependent dual meshes. Numerical results are given to illustrate the theoretical results.

References

A. B. Andreev and Raičo D. Lazarov, Superconvergence of the gradient for quadratic triangular finite elements, Numer. Methods Partial Differential Equations 4 (1988), no. 1, 15–32. MR 1012472, DOI 10.1002/num.1690040103
Ivo Babuška, Uday Banerjee, and John E. Osborn, Superconvergence in the generalized finite element method, Numer. Math. 107 (2007), no. 3, 353–395. MR 2336112, DOI 10.1007/s00211-007-0096-8
Randolph E. Bank and Donald J. Rose, Some error estimates for the box method, SIAM J. Numer. Anal. 24 (1987), no. 4, 777–787. MR 899703, DOI 10.1137/0724050
Randolph E. Bank and Jinchao Xu, Asymptotically exact a posteriori error estimators. I. Grids with superconvergence, SIAM J. Numer. Anal. 41 (2003), no. 6, 2294–2312. MR 2034616, DOI 10.1137/S003614290139874X
Erwin Stein, René de Borst, and Thomas J. R. Hughes (eds.), Encyclopedia of computational mechanics. Vol. 3, John Wiley & Sons, Ltd., Chichester, 2004. Fluids. MR 2288277, DOI 10.1002/0470091355
Zhi Qiang Cai, On the finite volume element method, Numer. Math. 58 (1991), no. 7, 713–735. MR 1090257, DOI 10.1007/BF01385651
Zhi Qiang Cai, Jan Mandel, and Steve McCormick, The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal. 28 (1991), no. 2, 392–402. MR 1087511, DOI 10.1137/0728022
Waixiang Cao, Zhimin Zhang, and Qingsong Zou, Superconvergence of any order finite volume schemes for 1D general elliptic equations, J. Sci. Comput. 56 (2013), no. 3, 566–590. MR 3081662, DOI 10.1007/s10915-013-9691-2
Waixiang Cao, Zhimin Zhang, and Qingsong Zou, Is $2k$-conjecture valid for finite volume methods?, SIAM J. Numer. Anal. 53 (2015), no. 2, 942–962. MR 3328149, DOI 10.1137/130936178
C. Chen and Y. Huang, High accuracy theory of finite element methods (in Chinese), Hunan Science and Technology Publishing House, Changsha, China, 1995.
Long Chen, A new class of high order finite volume methods for second order elliptic equations, SIAM J. Numer. Anal. 47 (2010), no. 6, 4021–4043. MR 2585177, DOI 10.1137/080720164
Z. Chen. L2 estimates of linear element generalized difference schemes. Acta Sci. Nat. Univ. Sunyatseni, 33:22–28, 1994.
Zhongying Chen, Ronghua Li, and Aihui Zhou, A note on the optimal $L^2$-estimate of the finite volume element method, Adv. Comput. Math. 16 (2002), no. 4, 291–303. MR 1894926, DOI 10.1023/A:1014577215948
Zhongying Chen, Junfeng Wu, and Yuesheng Xu, Higher-order finite volume methods for elliptic boundary value problems, Adv. Comput. Math. 37 (2012), no. 2, 191–253. MR 2944051, DOI 10.1007/s10444-011-9201-8
Zhongying Chen, Yuesheng Xu, and Yuanyuan Zhang, A construction of higher-order finite volume methods, Math. Comp. 84 (2015), no. 292, 599–628. MR 3290957, DOI 10.1090/S0025-5718-2014-02881-0
So-Hsiang Chou and Xiu Ye, Superconvergence of finite volume methods for the second order elliptic problem, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3706–3712. MR 2339996, DOI 10.1016/j.cma.2006.10.025
Bernardo Cockburn and Guosheng Fu, Superconvergence by $M$-decompositions. Part II: Construction of two-dimensional finite elements, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 1, 165–186. MR 3601005, DOI 10.1051/m2an/2016016
Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
Richard E. Ewing, Tao Lin, and Yanping Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal. 39 (2002), no. 6, 1865–1888. MR 1897941, DOI 10.1137/S0036142900368873
W. Hackbusch, On first and second order box schemes, Computing 41 (1989), no. 4, 277–296 (English, with German summary). MR 993825, DOI 10.1007/BF02241218
Jianguo Huang and Shitong Xi, On the finite volume element method for general self-adjoint elliptic problems, SIAM J. Numer. Anal. 35 (1998), no. 5, 1762–1774. MR 1640017, DOI 10.1137/S0036142994264699
Michal Křížek and Pekka Neittaanmäki, On superconvergence techniques, Acta Appl. Math. 9 (1987), no. 3, 175–198. MR 900263, DOI 10.1007/BF00047538
Ronghua Li, Zhongying Chen, and Wei Wu, Generalized difference methods for differential equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 226, Marcel Dekker, Inc., New York, 2000. Numerical analysis of finite volume methods. MR 1731376
Yong-hai Li and Rong-hua Li, Generalized difference methods on arbitrary quadrilateral networks, J. Comput. Math. 17 (1999), no. 6, 653–672. MR 1723103
F. Liebau, The finite volume element method with quadratic basis functions, Computing 57 (1996), no. 4, 281–299 (English, with English and German summaries). MR 1422087, DOI 10.1007/BF02252250
Q. Lin and N. Yan, The construction and analysis for efficient finite elements (in Chinese), Hebei Univ. Publ. House, 1996.
Runchang Lin and Zhimin Zhang, Natural superconvergence points in three-dimensional finite elements, SIAM J. Numer. Anal. 46 (2008), no. 3, 1281–1297. MR 2390994, DOI 10.1137/070681168
Tao Lin and Xiu Ye, A posteriori error estimates for finite volume method based on bilinear trial functions for the elliptic equation, J. Comput. Appl. Math. 254 (2013), 185–191. MR 3061075, DOI 10.1016/j.cam.2013.03.007
Yanping Lin, Min Yang, and Qingsong Zou, $L^2$ error estimates for a class of any order finite volume schemes over quadrilateral meshes, SIAM J. Numer. Anal. 53 (2015), no. 4, 2009–2029. MR 3384836, DOI 10.1137/140963121
Junliang Lv and Yonghai Li, $L^2$ error estimate of the finite volume element methods on quadrilateral meshes, Adv. Comput. Math. 33 (2010), no. 2, 129–148. MR 2659583, DOI 10.1007/s10444-009-9121-z
Junliang Lv and Yonghai Li, $L^2$ error estimates and superconvergence of the finite volume element methods on quadrilateral meshes, Adv. Comput. Math. 37 (2012), no. 3, 393–416. MR 2970858, DOI 10.1007/s10444-011-9215-2
Junliang Lv and Yonghai Li, Optimal biquadratic finite volume element methods on quadrilateral meshes, SIAM J. Numer. Anal. 50 (2012), no. 5, 2379–2399. MR 3022223, DOI 10.1137/100805881
Michael Plexousakis and Georgios E. Zouraris, On the construction and analysis of high order locally conservative finite volume-type methods for one-dimensional elliptic problems, SIAM J. Numer. Anal. 42 (2004), no. 3, 1226–1260. MR 2113684, DOI 10.1137/S0036142902406302
A. H. Schatz, I. H. Sloan, and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33 (1996), no. 2, 505–521. MR 1388486, DOI 10.1137/0733027
T. Schmidt, Box schemes on quadrilateral meshes, Computing 51 (1993), no. 3-4, 271–292 (English, with English and German summaries). MR 1253406, DOI 10.1007/BF02238536
Vidar Thomée, High order local approximations to derivatives in the finite element method, Math. Comp. 31 (1977), no. 139, 652–660. MR 438664, DOI 10.1090/S0025-5718-1977-0438664-4
Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050, DOI 10.1007/BFb0096835
Junping Wang and Xiu Ye, Superconvergence of finite element approximations for the Stokes problem by projection methods, SIAM J. Numer. Anal. 39 (2001), no. 3, 1001–1013. MR 1860454, DOI 10.1137/S003614290037589X
Xiang Wang, Weizhang Huang, and Yonghai Li, Conditioning of the finite volume element method for diffusion problems with general simplicial meshes, Math. Comp. 88 (2019), no. 320, 2665–2696. MR 3985472, DOI 10.1090/mcom/3423
Xiang Wang and Yonghai Li, $L^2$ error estimates for high order finite volume methods on triangular meshes, SIAM J. Numer. Anal. 54 (2016), no. 5, 2729–2749. MR 3544655, DOI 10.1137/140988486
Xiang Wang and Yonghai Li, Superconvergence of quadratic finite volume method on triangular meshes, J. Comput. Appl. Math. 348 (2019), 181–199. MR 3886669, DOI 10.1016/j.cam.2018.08.025
Xiang Wang and Yuqing Zhang, On the construction and analysis of finite volume element schemes with optimal $L^2$ convergence rate, Numer. Math. Theory Methods Appl. 14 (2021), no. 1, 47–70. MR 4161564, DOI 10.4208/nmtma.oa-2020-0027
Jinchao Xu and Qingsong Zou, Analysis of linear and quadratic simplicial finite volume methods for elliptic equations, Numer. Math. 111 (2009), no. 3, 469–492. MR 2470148, DOI 10.1007/s00211-008-0189-z
Min Yang, Chunjia Bi, and Jiangguo Liu, Postprocessing of a finite volume element method for semilinear parabolic problems, M2AN Math. Model. Numer. Anal. 43 (2009), no. 5, 957–971. MR 2559740, DOI 10.1051/m2an/2009017
Zhimin Zhang, Superconvergence points of polynomial spectral interpolation, SIAM J. Numer. Anal. 50 (2012), no. 6, 2966–2985. MR 3022250, DOI 10.1137/120861291
Zhimin Zhang and Ahmed Naga, A new finite element gradient recovery method: superconvergence property, SIAM J. Sci. Comput. 26 (2005), no. 4, 1192–1213. MR 2143481, DOI 10.1137/S1064827503402837
Zhimin Zhang and Qingsong Zou, A family of finite volume schemes of arbitrary order on rectangular meshes, J. Sci. Comput. 58 (2014), no. 2, 308–330. MR 3150261, DOI 10.1007/s10915-013-9737-5
Zhimin Zhang and Qingsong Zou, Vertex-centered finite volume schemes of any order over quadrilateral meshes for elliptic boundary value problems, Numer. Math. 130 (2015), no. 2, 363–393. MR 3343929, DOI 10.1007/s00211-014-0664-7
Q. Zhu and Q. Lin. The superconvergence theory of finite elements (In Chinese). Hunan Science and Technology Publishing House, Changsha, China, 1989.
O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331–1364. MR 1161557, DOI 10.1002/nme.1620330702