| Title:
|
The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type (English) |
| Author:
|
Zhang, Tie |
| Author:
|
Zhang, Shuhua |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
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60 |
| Issue:
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5 |
| Year:
|
2015 |
| Pages:
|
573-596 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$-uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$, the gradient approximation possesses the superconvergence: $\max \nolimits _{P\in S}|(\nabla u-\overline {\nabla }u_h)(P)|=O(h^2)\mathopen |\ln h|^{{3}/{2}}$, where $\overline {\nabla }$ denotes the average gradient on elements containing vertex $P$. Furthermore, by using the interpolation post-processing technique, we also derive a global superconvergence estimate in the $H^1$-norm and establish an asymptotically exact a posteriori error estimator for the error $\|u-u_h\|_1$. (English) |
| Keyword:
|
finite volume method |
| Keyword:
|
nonlinear elliptic problem |
| Keyword:
|
local and global superconvergence in the $W^{1,\infty }$-norm |
| Keyword:
|
a posteriori error estimator |
| MSC:
|
65M15 |
| MSC:
|
65M60 |
| idZBL:
|
Zbl 06486926 |
| idMR:
|
MR3396481 |
| DOI:
|
10.1007/s10492-015-0112-8 |
| . |
| Date available:
|
2015-09-03T10:46:34Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144392 |
| . |
| Reference:
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[1] Babuška, I., Banerjee, U., Osborn, J. E.: Superconvergence in the generalized finite element method.Numer. Math. 107 (2007), 353-395. Zbl 1129.65075, MR 2336112, 10.1007/s00211-007-0096-8 |
| Reference:
|
[2] Babuška, I., Strouboulis, T., Upadhyay, C. S., Gangaraj, S. K.: Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations.Numer. Methods Partial Differ. Equations 12 (1996), 347-392. Zbl 0854.65089, MR 1388445, 10.1002/num.1690120303 |
| Reference:
|
[3] Babuška, I., Whiteman, J. R., Strouboulis, T.: Finite Elements. An Introduction to the Method and Error Estimation.Oxford University Press, Oxford (2011). Zbl 1206.65246, MR 2857237 |
| Reference:
|
[4] Bank, R. E., Rose, D. J.: Some error estimates for the box method.SIAM J. Numer. Anal. 24 (1987), 777-787. Zbl 0634.65105, MR 0899703, 10.1137/0724050 |
| Reference:
|
[5] Bergam, A., Mghazli, Z., Verfürth, R.: A posteriori estimates of a finite volume scheme for a nonlinear problem.Numer. Math. French 95 (2003), 599-624. Zbl 1033.65095, MR 2013121 |
| Reference:
|
[6] Bi, C.: Superconvergence of finite volume element method for a nonlinear elliptic problem.Numer. Methods Partial Differ. Equations 23 (2007), 220-233. Zbl 1119.65105, MR 2275467, 10.1002/num.20173 |
| Reference:
|
[7] Bi, C., Ginting, V.: Two-grid finite volume element method for linear and nonlinear elliptic problems.Numer. Math. 108 (2007), 177-198. Zbl 1134.65077, MR 2358002, 10.1007/s00211-007-0115-9 |
| Reference:
|
[8] Brandts, J. H.: Analysis of a non-standard mixed finite element method with applications to superconvergence.Appl. Math., Praha 54 (2009), 225-235. Zbl 1212.65441, MR 2530540, 10.1007/s10492-009-0014-8 |
| Reference:
|
[9] Brandts, J., Křížek, M.: Gradient superconvergence on uniform simplicial partitions of polytopes.IMA J. Numer. Anal. 23 (2003), 489-505. Zbl 1042.65081, MR 1987941, 10.1093/imanum/23.3.489 |
| Reference:
|
[10] Brandts, J., Křížek, M.: Superconvergence of tetrahedral quadratic finite elements.J. Comput. Math. 23 (2005), 27-36. Zbl 1072.65137, MR 2124141 |
| Reference:
|
[11] Cai, Z.: On the finite volume element method.Numer. Math. 58 (1991), 713-735. Zbl 0731.65093, MR 1090257, 10.1007/BF01385651 |
| Reference:
|
[12] Chatzipantelidis, P., Ginting, V., Lazarov, R. D.: A finite volume element method for a nonlinear elliptic problem.Numer. Linear Algebra Appl. 12 (2005), 515-546. MR 2150166, 10.1002/nla.439 |
| Reference:
|
[13] Chen, Z.: Superconvergence of generalized difference method for elliptic boundary value problem.Numer. Math., J. Chin. Univ. 3 (1994), 163-171. Zbl 0814.65102, MR 1325662 |
| Reference:
|
[14] Chen, L.: A new class of high order finite volume methods for second order elliptic equations.SIAM J. Numer. Anal. 47 (2010), 4021-4043. Zbl 1261.65109, MR 2585177, 10.1137/080720164 |
| Reference:
|
[15] Chen, Z., Li, R., Zhou, A.: A note on the optimal $L^2$-estimate of the finite volume element method.Adv. Comput. Math. 16 (2002), 291-303. Zbl 0997.65122, MR 1894926, 10.1023/A:1014577215948 |
| Reference:
|
[16] Chou, S.-H., Kwak, D. Y., Li, Q.: $L^p$ error estimates and superconvergence for covolume or finite volume element methods.Numer. Methods Partial Differ. Equations 19 (2003), 463-486. Zbl 1029.65123, MR 1980190, 10.1002/num.10059 |
| Reference:
|
[17] J. Douglas, Jr., T. Dupont: A Galerkin method for a nonlinear Dirichlet problem.Math. Comp. 29 (1975), 689-696. Zbl 0306.65072, MR 0431747, 10.1090/S0025-5718-1975-0431747-2 |
| Reference:
|
[18] J. Douglas, Jr., T. Dupont, J. Serrin: Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form.Arch Ration. Mech. Anal. 42 (1971), 157-168. Zbl 0222.35017, MR 0393829, 10.1007/BF00250482 |
| Reference:
|
[19] Ewing, R. E., Lin, T., Lin, Y.: On the accuracy of the finite volume element method based on piecewise linear polynomials.SIAM J. Numer. Anal. 39 (2002), 1865-1888. Zbl 1036.65084, MR 1897941, 10.1137/S0036142900368873 |
| Reference:
|
[20] Hlaváček, I., Křížek, M.: On a nonpotential nonmonotone second order elliptic problem with mixed boundary conditions.Stab. Appl. Anal. Contin. Media 3 (1993), 85-97. |
| Reference:
|
[21] Hlaváček, I., Křížek, M., Malý, J.: On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type.J. Math. Anal. Appl. 184 (1994), 168-189. MR 1275952, 10.1006/jmaa.1994.1192 |
| Reference:
|
[22] Huang, J., Li, L.: Some superconvergence results for the covolume method for elliptic problems.Commun. Numer. Methods Eng. 17 (2001), 291-302. Zbl 0987.65109, MR 1832578, 10.1002/cnm.403 |
| Reference:
|
[23] Křížek, M., Neittaanmäki, P.: Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications.Mathematical Modelling: Theory and Applications 1 Kluwer Academic Publishers, Dordrecht (1996). MR 1431889 |
| Reference:
|
[24] Lazarov, R. D., Mishev, I. D., Vassilevski, P. S.: Finite volume methods for convection-diffusion problems.SIAM J. Numer. Anal. 33 (1996), 31-55. Zbl 0847.65075, MR 1377242, 10.1137/0733003 |
| Reference:
|
[25] Li, R.: Generalized difference methods for a nonlinear Dirichlet problem.SIAM J. Numer. Anal. 24 (1987), 77-88. Zbl 0626.65091, MR 0874736, 10.1137/0724007 |
| Reference:
|
[26] Li, R., Chen, Z., Wu, W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods.Pure and Applied Mathematics Marcel Dekker, New York (2000). Zbl 0940.65125, MR 1731376 |
| Reference:
|
[27] Lin, Q., Zhu, Q. D.: The Preprocessing and Postprocessing for the Finite Element Methods.Chinese Shanghai Sci. & Tech. Publishing Shanghai (1994). |
| Reference:
|
[28] Lv, J., Li, Y.: $L^2$ error estimates and superconvergence of the finite volume element methods on quadrilateral meshes.Adv. Comput. Math. 37 (2012), 393-416. Zbl 1255.65198, MR 2970858, 10.1007/s10444-011-9215-2 |
| Reference:
|
[29] Schmidt, T.: Box schemes on quadrilateral meshes.Computing 51 (1993), 271-292. Zbl 0787.65075, MR 1253406, 10.1007/BF02238536 |
| Reference:
|
[30] Süli, E.: Convergence of finite volume schemes for Poisson's equation on nonuniform meshes.SIAM J. Numer. Anal. 28 (1991), 1419-1430. Zbl 0802.65104, MR 1119276, 10.1137/0728073 |
| Reference:
|
[31] Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods.Lecture Notes in Mathematics 1605 Springer, Belin (1995). Zbl 0826.65092, MR 1439050, 10.1007/BFb0096835 |
| Reference:
|
[32] Wu, H., Li, R.: Error estimates for finite volume element methods for general second-order elliptic problems.Numer. Methods Partial Differ. Equations 19 (2003), 693-708. Zbl 1040.65091, MR 2009589, 10.1002/num.10068 |
| Reference:
|
[33] Zhang, T.: Finite Element Methods for Partial Differential-Integral Equations.Chinese Science Press, Beijing (2012). |
| Reference:
|
[34] Zhang, T., Lin, Y. P., Tait, R. J.: On the finite volume element version of Ritz-Volterra projection and applications to related equations.J. Comput. Math. 20 (2002), 491-504. Zbl 1013.65143, MR 1931591 |
| Reference:
|
[35] Zhu, Q. D., Lin, Q.: The Superconvergence Theory of Finite Elements.Chinese Hunan Science and Technology Publishing House Changsha (1989). MR 1200243 |
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