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least-squares finite element method; mixed finite element method; natural superconvergence; Raviart-Thomas element; Poisson equation; Lagrange elements; triangular and rectangular meshes; numerical experiments; Galerkin method
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.
[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
[2] Babuška, I.: Error bounds for finite element method. Numer. Math. 16 (1971), 322-333. DOI 10.1007/BF02165003 | MR 0288971
[3] Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability. Clarendon Press Oxford (2001). MR 1857191
[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. DOI 10.1002/num.1690120303 | MR 1388445
[5] Bedivan, D. M.: Error estimates for least squares finite element methods. Comput. Math. Appl. 43 (2002), 1003-1020. DOI 10.1016/S0898-1221(02)80009-8 | MR 1892481 | Zbl 1050.65098
[6] Bochev, P. B., Gunzburger, M. D.: Finite element methods of least-squares type. SIAM Rev. 40 (1998), 789-837. DOI 10.1137/S0036144597321156 | MR 1659689 | Zbl 0914.65108
[7] Brandts, J. H.: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68 (1994), 311-324. DOI 10.1007/s002110050064 | MR 1313147 | Zbl 0823.65103
[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. DOI 10.1016/S0168-9274(99)00034-3 | MR 1755693 | Zbl 0948.65120
[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
[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. DOI 10.1093/imanum/dri048 | MR 2269196 | Zbl 1106.65102
[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. MR 0365287 | Zbl 0338.90047
[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
[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. DOI 10.1007/BF01389710 | MR 0799685 | Zbl 0599.65072
[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. DOI 10.1137/050636504 | MR 2257124 | Zbl 1138.76053
[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. DOI 10.1137/0731091 | MR 1302685 | Zbl 0813.65119
[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. DOI 10.1002/cnm.1630050702 | Zbl 0684.65083
[17] Chen, C. M.: Structure Theory of Superconvergence of Finite Elements. Hunan Science Press Hunan (2001), Chinese.
[18] Chen, C. M., Huang, Y. Q.: High Accuracy Theory of Finite Element Methods. Hunan Science and Technology Press Hunan (1995), Chinese.
[19] Chen, Y.: Superconvergence of mixed finite element methods for optimal control problems. Math. Comput. 77 (2008), 1269-1291. DOI 10.1090/S0025-5718-08-02104-2 | MR 2398768 | Zbl 1193.49029
[20] Chen, Z.: Finite Element Methods and Their Applications. Scientific Computation. Springer Berlin (2005).
[21] Douglas, J., Dupont, T.: Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces. Numer. Math. 22 (1974), 99-109. DOI 10.1007/BF01436724 | MR 0362922 | Zbl 0331.65051
[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
[23] Douglas, J., Wang, J.: Superconvergence of mixed finite element methods on rectangular domains. Calcolo 26 (1989), 121-133. DOI 10.1007/BF02575724 | MR 1083049 | Zbl 0714.65084
[24] Durán, R.: Superconvergence for rectangular mixed finite elements. Numer. Math. 58 (1990), 287-298. DOI 10.1007/BF01385626 | MR 1075159
[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. DOI 10.1137/0728054 | MR 1111451 | Zbl 0733.65065
[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. DOI 10.1137/S0036142997322801 | MR 1681041 | Zbl 0926.65107
[27] Ewing, R. E., Wang, J.: Analysis of mixed finite element methods on locally refined grids. Numer. Math. 63 (1992), 183-194. DOI 10.1007/BF01385855 | MR 1182973 | Zbl 0772.65071
[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. DOI 10.1007/BF01408578 | MR 0880337
[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. MR 1015921 | Zbl 0673.65060
[30] Jiang, B.-N.: The Least-Squares Finite Element Method. Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer Berlin (1998). MR 1639101
[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
[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
[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. DOI 10.1137/S0036142999351212 | MR 1781203 | Zbl 0974.65106
[34] Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. Science Press Beijing (2006).
[35] Lin, Q., Pan, J. H.: High accuracy for mixed finite element methods in Raviart-Thomas element. J. Comput. Math. 14 (1996), 175-182. MR 1399911 | Zbl 0846.65062
[36] Lin, Q., Yan, N.: Construction and Analysis of High Efficient Finite Elements. Hebei University Press Hebei (1996), Chinese.
[37] Lin, R., Zhang, Z.: Natural superconvergent points of triangular finite elements. Numer. Methods Partial Differ. Equations 20 (2004), 864-906. DOI 10.1002/num.20013 | MR 2092411 | Zbl 1068.65123
[38] Lin, R., Zhang, Z.: Convergence analysis for least-squares approximations to solutions of second-order two-point boundary value problems. Submitted.
[39] Pehlivanov, A. I., Carey, G. F.: Error estimates for least-squares mixed finite elements. RAIRO, Modélisation Math. Anal. Numér. 28 (1994), 499-516. MR 1295584 | Zbl 0820.65065
[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. DOI 10.1137/0731071 | MR 1293520 | Zbl 0806.65108
[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. DOI 10.1007/BF02243846 | MR 1248894 | Zbl 0790.65079
[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
[43] Verfürth, R.: A Review of Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner Chichester-Stuttgart (1996).
[44] Wahlbin, L. B.: Superconvergence in Galerkin Finite Flement Methods. Lecture Notes Math. 1605. Springer Berlin (1995). MR 1439050
[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. DOI 10.1137/0710077 | MR 0343659 | Zbl 0266.65061
[46] Yan, N.: Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods. Science Press Beijing (2008).
[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. DOI 10.1090/S0025-5718-98-00942-9 | MR 1459393
[48] Zhang, Z.: Recovery techniques in finite element methods. In: Adaptive Computations: Theory and Algorithms T. Tang, J. Xu Science Publisher (2007), 297-365.
[49] Zhu, Q.: High Accuracy and Post-Processing Theory of the Finite Element Method. Science Press Beijing (2008), Chinese.
[50] Zienkiewicz, O. C., Taylor, R. L., Zhu, J. Z.: The Finite Element Method, 6th ed. McGraw-Hill London (2005).
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