| Title:
|
Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems (English) |
| Author:
|
Lin, Runchang |
| Author:
|
Zhang, Zhimin |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
54 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
251-266 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method. (English) |
| Keyword:
|
least-squares finite element method |
| Keyword:
|
mixed finite element method |
| Keyword:
|
natural superconvergence |
| Keyword:
|
Raviart-Thomas element |
| Keyword:
|
Poisson equation |
| Keyword:
|
Lagrange elements |
| Keyword:
|
triangular and rectangular meshes |
| Keyword:
|
numerical experiments |
| Keyword:
|
Galerkin method |
| MSC:
|
35J05 |
| MSC:
|
65N12 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 1212.65419 |
| idMR:
|
MR2530542 |
| DOI:
|
10.1007/s10492-009-0016-6 |
| . |
| Date available:
|
2010-07-20T13:04:07Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140363 |
| . |
| Reference:
|
[1] Ainsworth, M., Oden, J. T.: A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics.Wiley Interscience, John Wiley & Sons New York (2000). MR 1885308 |
| Reference:
|
[2] Babuška, I.: Error bounds for finite element method.Numer. Math. 16 (1971), 322-333. MR 0288971, 10.1007/BF02165003 |
| Reference:
|
[3] Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability.Clarendon Press Oxford (2001). MR 1857191 |
| Reference:
|
[4] 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, Poisson, and the elasticity equations.Numer. Methods Partial Differ. Equations 12 (1996), 347-392. MR 1388445, 10.1002/num.1690120303 |
| Reference:
|
[5] Bedivan, D. M.: Error estimates for least squares finite element methods.Comput. Math. Appl. 43 (2002), 1003-1020. Zbl 1050.65098, MR 1892481, 10.1016/S0898-1221(02)80009-8 |
| Reference:
|
[6] Bochev, P. B., Gunzburger, M. D.: Finite element methods of least-squares type.SIAM Rev. 40 (1998), 789-837. Zbl 0914.65108, MR 1659689, 10.1137/S0036144597321156 |
| Reference:
|
[7] Brandts, J. H.: Superconvergence and a posteriori error estimation for triangular mixed finite elements.Numer. Math. 68 (1994), 311-324. Zbl 0823.65103, MR 1313147, 10.1007/s002110050064 |
| Reference:
|
[8] Brandts, J. H.: Superconvergence for triangular order $k=1$ Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods.Appl. Numer. Math. 34 (2000), 39-58. Zbl 0948.65120, MR 1755693, 10.1016/S0168-9274(99)00034-3 |
| Reference:
|
[9] Brandts, J. H., Chen, Y. P.: Superconvergence of least-squares mixed finite element methods.Int. J. Numer. Anal. Model. 3 (2006), 303-311. MR 2237884 |
| Reference:
|
[10] Brandts, J. H., Chen, Y. P., Yang, J.: A note on least-squares mixed finite elements in relation to standard and mixed finite elements.IMA J. Numer. Anal. 26 (2006), 779-789. Zbl 1106.65102, MR 2269196, 10.1093/imanum/dri048 |
| Reference:
|
[11] Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers.Rev. Franc. Automat. Inform. Rech. Operat. 8 (1974), 129-151. Zbl 0338.90047, MR 0365287 |
| Reference:
|
[12] Brezzi, F., J. Douglas, Jr., Fortin, M., Marini, L. D.: Efficient rectangular mixed finite elements in two and three space variables.Mathematical Modelling and Numerical Analysis 21 (1987), 581-604. MR 0921828, 10.1051/m2an/1987210405811 |
| Reference:
|
[13] Brezzi, F., J. Douglas, Jr., Marini, L. D.: Two families of mixed finite elements for second order elliptic problems.Numer. Math. 47 (1985), 217-235. Zbl 0599.65072, MR 0799685, 10.1007/BF01389710 |
| Reference:
|
[14] Cai, Z., Ku, J.: The $L^2$ norm error estimates for the div least-squares method.SIAM J. Numer. Anal. 44 (2006), 1721-1734. Zbl 1138.76053, MR 2257124, 10.1137/050636504 |
| Reference:
|
[15] Cai, Z., Lazarov, R. D., Manteuffel, T. A., McCormick, S. F.: First-order system least squares for second-order partial differential equations. I.SIAM J. Numer. Anal. 31 (1994), 1785-1799. Zbl 0813.65119, MR 1302685, 10.1137/0731091 |
| Reference:
|
[16] Carey, G. F., Shen, Y.: Convergence studies of least-squares finite elements for first-order systems.Commun. Appl. Numer. Methods 5 (1989), 427-434. Zbl 0684.65083, 10.1002/cnm.1630050702 |
| Reference:
|
[17] Chen, C. M.: Structure Theory of Superconvergence of Finite Elements.Hunan Science Press Hunan (2001), Chinese. MR 0840307 |
| Reference:
|
[18] Chen, C. M., Huang, Y. Q.: High Accuracy Theory of Finite Element Methods.Hunan Science and Technology Press Hunan (1995), Chinese. |
| Reference:
|
[19] Chen, Y.: Superconvergence of mixed finite element methods for optimal control problems.Math. Comput. 77 (2008), 1269-1291. Zbl 1193.49029, MR 2398768, 10.1090/S0025-5718-08-02104-2 |
| Reference:
|
[20] Chen, Z.: Finite Element Methods and Their Applications. Scientific Computation.Springer Berlin (2005). MR 2158541 |
| Reference:
|
[21] Douglas, J., Dupont, T.: Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces.Numer. Math. 22 (1974), 99-109. Zbl 0331.65051, MR 0362922, 10.1007/BF01436724 |
| Reference:
|
[22] Douglas, J., Dupont, T., Wahlbin, L.: Optimal $L^\infty$ error estimates for Galerkin approximations to solutions of two-point boundary value problems.Math. Comput. 29 (1975), 475-483. MR 0371077 |
| Reference:
|
[23] Douglas, J., Wang, J.: Superconvergence of mixed finite element methods on rectangular domains.Calcolo 26 (1989), 121-133. Zbl 0714.65084, MR 1083049, 10.1007/BF02575724 |
| Reference:
|
[24] Durán, R.: Superconvergence for rectangular mixed finite elements.Numer. Math. 58 (1990), 287-298. MR 1075159, 10.1007/BF01385626 |
| Reference:
|
[25] Ewing, R. E., Lazarov, R. D., Wang, J.: Superconvergence of the velocity along the Gauss lines in mixed finite element methods.SIAM J. Numer. Anal. 28 (1991), 1015-1029. Zbl 0733.65065, MR 1111451, 10.1137/0728054 |
| Reference:
|
[26] Ewing, R. E., Liu, M. M., Wang, J.: Superconvergence of mixed finite element approximations over quadrilaterals.SIAM J. Numer. Anal. 36 (1998), 772-787. Zbl 0926.65107, MR 1681041, 10.1137/S0036142997322801 |
| Reference:
|
[27] Ewing, R. E., Wang, J.: Analysis of mixed finite element methods on locally refined grids.Numer. Math. 63 (1992), 183-194. Zbl 0772.65071, MR 1182973, 10.1007/BF01385855 |
| Reference:
|
[28] Gastaldi, L., Nochetto, R. H.: Optimal $L^\infty$-error estimates for nonconforming and mixed finite element methods of lowest order.Numer. Math. 50 (1987), 587-611. MR 0880337, 10.1007/BF01408578 |
| Reference:
|
[29] Gastaldi, L., Nochetto, R. H.: Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations.RAIRO, Modélisation Math. Anal. Numér. 23 (1989), 103-128. Zbl 0673.65060, MR 1015921, 10.1051/m2an/1989230101031 |
| Reference:
|
[30] Jiang, B.-N.: The Least-Squares Finite Element Method. Theory and Applications in Computational Fluid Dynamics and Electromagnetics.Springer Berlin (1998). MR 1639101 |
| Reference:
|
[31] Křížek, M., Neittaanmäki, P.: Bibliography on superconvergence.Finite element methods. Superconvergence, post-processing, and a posteriori estimates M. Kř'ižek, P. Neittaanmäki, R. Stenberg Marcel Dekker New York (1998), 315-348. MR 1602730 |
| Reference:
|
[32] Křížek, M., Neittaanmäki, P.: Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, 50.Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1990). MR 1066462 |
| Reference:
|
[33] Li, J., Wheeler, M. F.: Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids.SIAM J. Numer. Anal. 38 (2000), 770-798. Zbl 0974.65106, MR 1781203, 10.1137/S0036142999351212 |
| Reference:
|
[34] Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement.Science Press Beijing (2006). |
| Reference:
|
[35] Lin, Q., Pan, J. H.: High accuracy for mixed finite element methods in Raviart-Thomas element.J. Comput. Math. 14 (1996), 175-182. Zbl 0846.65062, MR 1399911 |
| Reference:
|
[36] Lin, Q., Yan, N.: Construction and Analysis of High Efficient Finite Elements.Hebei University Press Hebei (1996), Chinese. |
| Reference:
|
[37] Lin, R., Zhang, Z.: Natural superconvergent points of triangular finite elements.Numer. Methods Partial Differ. Equations 20 (2004), 864-906. Zbl 1068.65123, MR 2092411, 10.1002/num.20013 |
| Reference:
|
[38] Lin, R., Zhang, Z.: Convergence analysis for least-squares approximations to solutions of second-order two-point boundary value problems.Submitted. |
| Reference:
|
[39] Pehlivanov, A. I., Carey, G. F.: Error estimates for least-squares mixed finite elements.RAIRO, Modélisation Math. Anal. Numér. 28 (1994), 499-516. Zbl 0820.65065, MR 1295584, 10.1051/m2an/1994280504991 |
| Reference:
|
[40] Pehlivanov, A. I., Carey, G. F., Lazarov, R. D.: Least-squares mixed finite elements for second-order elliptic problems.SIAM J. Numer. Anal. 31 (1994), 1368-1377. Zbl 0806.65108, MR 1293520, 10.1137/0731071 |
| Reference:
|
[41] Pehlivanov, A. I., Carey, G. F., Lazarov, R. D., Shen, Y.: Convergence analysis of least-squares mixed finite elements.Computing 51 (1993), 111-123. Zbl 0790.65079, MR 1248894, 10.1007/BF02243846 |
| Reference:
|
[42] Raviart, P. A., Thomas, J. M.: A mixed finite element method for second order elliptic problems.In: Mathematical Aspects of the Finite Element Method. Lecture Notes Math. 606 I. Galligani, E. Magenes Springer Berlin (1977), 292-315. MR 0483555 |
| Reference:
|
[43] Verfürth, R.: A Review of Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques.Wiley-Teubner Chichester-Stuttgart (1996). |
| Reference:
|
[44] Wahlbin, L. B.: Superconvergence in Galerkin Finite Flement Methods. Lecture Notes Math. 1605.Springer Berlin (1995). MR 1439050 |
| Reference:
|
[45] Wheeler, M. F.: An optimal $L_{\infty}$ error estimate for Galerkin approximations to solutions of two-point boundary value problems.SIAM J. Numer. Anal. 10 (1973), 914-917. Zbl 0266.65061, MR 0343659, 10.1137/0710077 |
| Reference:
|
[46] Yan, N.: Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods.Science Press Beijing (2008). |
| Reference:
|
[47] Zhang, Z.: Derivative superconvergence points in finite element solutions of Poisson equation for the serendipity and intermediate families. A theoretical justification.Math. Comput. 67 (1998), 541-552. MR 1459393, 10.1090/S0025-5718-98-00942-9 |
| Reference:
|
[48] Zhang, Z.: Recovery techniques in finite element methods.In: Adaptive Computations: Theory and Algorithms T. Tang, J. Xu Science Publisher (2007), 297-365. |
| Reference:
|
[49] Zhu, Q.: High Accuracy and Post-Processing Theory of the Finite Element Method.Science Press Beijing (2008), Chinese. |
| Reference:
|
[50] Zienkiewicz, O. C., Taylor, R. L., Zhu, J. Z.: The Finite Element Method, 6th ed.McGraw-Hill London (2005). MR 3292660 |
| . |
