Abstract
A brief survey with a bibliography of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented. A particular emphasis is laid on superconvergent schemes for elliptic problems in the plane employing the finite element method.
Similar content being viewed by others
References
Adeboye, K. R.: Superconvergence results for Galerkin method for parabolic initial value problems via Laplace transform, Bull. Math. Soc. Sci. Math. R.S. Roumanie (N.S.) 26 (1982), 115–127. MR 84h:65095.
Andreev, A. B.: Superconvergence of the gradient for linear triangle elements for elliptic and parabolic equations, C.R. Acad. Bulgare Sci. 37 (1984), 293–296. MR 85h:65230.
Andreev, A. B.: Error estimate of type superconvergence of the gradient for quadratic triangular elements, C.R. Acad. Bulgare Sci. 37 (1984), 1179–1182. MR 86f:65186.
Arnold, D. N. and Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), 7–32.
Arnold, D. N. and Douglas, J. J.: Superconvergence of the Galerkin approximation of a quasilinear parabolic equation in a single space variable, Calcolo 16 (1979), 345–369. MR 83b:65118.
Arnold, D. N., Douglas, J. J., and Thomée, V.: Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981), 53–63. MR 82f:65108.
Arnold, D. N. and Saranen, J.: On the asymptotic convergence of spline collocation methods for partial differential equations, SIAM J. Numer. Anal. 21 (1984), 459–472.
Arnold, D. N. and Wendland, W. L.: On the asymptotic convergence of collocation methods, Math. Comp. 41 (1983), 349–381. MR 85h:65254.
Arnold, D. N. and Wendland, W. L.: Collocation versus Galerkin procedures for boundary integral methods, in Boundary Element Methods in Engineering, Springer-Verlag, Berlin, 1982, pp. 18–33. MR 85f:73074.
Arnold, D. N. and Wendland, W. L.: The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math. 47 (1985), 317–341.
Arnold, D. N. and Winther, R.: A superconvergent finite element method for the Korteweg-de Vries equation. Math. Comp. 38 (1982), 23–36. MR 82m:65087.
Ascher, U.: Discrete least squares approximations for ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), 478–496. MR 81e:65043.
Ascher, U. and Weiss, R.: Collocation for singular perturbation. I. First order systems with constant coefficients, SIAM J. Numer. Anal. 20 (1983), 537–557. MR 85a:65113.
Babuška, I., Izadpanah, K., and Szabo, B.: The post-processing technique in the finite element method. The theory and experience, Technical Note BN-1016, Univ. of Maryland, 1984, 1–25.
Babuška, I. and Miller, A.: The post-processing in the finite element method, Parts I–II, Internat. J. Numer. Methods Engrg. 20 (1984), 1085–1109, 1111–1129.
Bakker, M.: On the numerical solution of parabolic equations in a single space variable by the continuous time Galerkin method, SIAM J. Numer. Anal. 17 (1980), 162–177. MR 81m:65167.
Bakker, M.: A note on C 0 Galerkin methods for two-point boundary problems, Numer. Math. 38 (1981/82), 447–453. MR 83e:65140.
Bakker, M.: One-dimensional Galerkin methods and superconvergence at interior nodal points, SIAM J. Numer. Anal. 21 (1984), 101–110. MR 85f:65080.
Barlow, J.: Optimal stress location in finite-element models, Internat. J. Numer. Methods Engrg. 10 (1976), 243–251.
Beatson, R.: Joint approximation of a function and its derivatives, in Approximation Theory III, (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York, 1980, pp. 199–206. MR 82c:41019.
Behforooz, G. H. and Papamichael, N.: Improved orders of approximation derived from interpolatory cubic splines. BIT 19 (1979), 19–26. MR 80e:41004.
Blum, H., Lin, Q., and Rannacher, R.: Asymptotic error expansion and Richardson extrapolation for linear finite elements, Numer. Math. 49 (1986), 11–38.
De Boor, C. and Swartz, B.: Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582–606. MR 51#9528.
De Boor, C. and Swartz, B.: Collocation approximation to eigenvalues of an ordinary differential equation: The principle of the thing, Math. Comp. 35 (1980), 679–694. MR 81k:65097.
De Boor, C. and Swartz, B.: Local piecewise polynomial projection methods for an ODE which give high-order convergence at knots, Math. Comp. 36 (1981), 21–33. MR 82f:65091.
Bramble, J. H. and Schatz, A. H.: Estimates for spline projection, RAIRO Anal. Numér. 10 (1976), 5–37. MR 55#9563.
Bramble, J. H. and Schatz, A. H.: Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), 94–111. MR 55#4739.
Bramble, J. H., and Thomée, V.: Interior maximum norm estimates for some simple finite element methods, RAIRO Anal. Numér. 8 (1974), 5–18. MR 50#11808.
Brunner, H.: On superconvergence in collocation methods for Abel integral equations, Numerical Mathematics and Computing (Proc. Eighth Manitoba Conf., Winnipeg, 1978). Utilitas Mathematica Publishing, Inc., Winnipeg, Man, 1979, pp. 117–128. MR 80m: 45023.
Brunner, H.: Superconvergence of collocation methods for Volterra integral equations of the first kind, Computing 21 (1979), 151–157. MR 83a:65125.
Brunner, H.: A note on collocation methods for Volterra integral equations of the first kind, Computing 23 (1979), 179–187. MR 83b:65145.
Brunner, H.: Superconvergence in collocation and implicit Runge-Kutta methods for Volterratype integral equations of the second kind, Numerical Treatment of Integral Equations (Work-shop, Math. Res. Inst., Oberwolfach, 1979), Internat. Ser. Numer. Math., 53, Birkhäuser, Basel 1980, pp. 54–72. MR 81m:65194.
Brunner, H.: The application of the variation of constants formulas in the numerical analysis of integral and integro-differential equations, Utilitas Math. 19 (1981), 255–290. MR 83b:65146.
Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations, SIAM J. Numer. Anal. 21 (1984), 1132–1145. MR 86d:65160.
Brunner, H. and Sørsett, S. P.: Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind, Numer. Math. 36 (1981), 347–358. MR 83e:65202.
Carey, G. F., Humphrey, D., and Wheeler, M. F.: Galerkin and collocation-Galerkin methods with superconvergence and optimal fluxes, Internat. J. Numer. Methods Engrg. 17 (1981), 939–950. MR 82g:80008.
Carey, G. F. and Oden, J. T.: Finite Elements. Vol. II: A second course, The Texas Finite Element Series, Vol. 2, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.
Carey, G. F. and Wheeler, M. F.: C 0-collocation-Galerkin methods, Codes for Boundary-Value Problems in Ordinary Differential Equations (Proc. Conf., Houston, 1978), LN in Comp. Sci., Vol. 76, Springer, Berlin, 1979, pp. 250–256. MR 82b:65074.
Chandler, G. A.: Global superconvergence of iterated Galerkin solutions for second kind integral equations, Technical Report, Australian Nat. Univ., Canberra, 1978.
40.Chandler, G. A.: Superconvergence of numerical solutions to second kind integral equations, PhD thesis, Australian Nat. Univ., Canberra, 1979.
Chandler, G. A.: Superconvergence for second kind integral equations, The Application and Numerical Solution of Integral Equations (Proc. Conf., Canberra, 1978), Sijthoff & Noordhoff, Alphen-aan-den-Rijn, 1980, pp. 103–117. MR 81h:45027.
Chandler, G. A.: Galerkin's method for boundary integral equations on polygonal domains, J. Austral. Math. Soc. Ser. B 26 (1984), 1–13, MR 86a:65116.
Chatelin, F.: Sur les bornes d'erreur a posteriori pour les éléments propres d'opérateurs linéaires, Numer. Math. 32 (1979), 233–246. MR 81d:65027.
Chatelin, F.: Des resultats de superconvergence, Seminaire d'analyse numérique, No. 331, Univ. Scientifique et Medicale de Grenoble, Laboratoire IMAG, 1980.
Chatelin, F.: The spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators, SIAM Rev. 23 (1981), 495–522. MR 83b:47023.
Chatelin, F.: Spectral Approximation of Linear Operators, Academic Press, New York. 1983.
Chatelin, F. and Lebbar, R.: The interated projection solution for the Fredholm integral equation of second kind, J. Austral. Math. Soc. Ser. B 22 (1981), 439–451. MR 82h:65096.
Chatelin, F. and Lebbar, R.: Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem, J. Integral Equations 6 (1984), 71–91. MR 85i:65167.
Chen, C. M.: The good points of the approximation solution for Galerkin method for two-point boundary problem, Numer. Math. J. Chinese Univ. 1 (1979), 73–79, MR 82e:65085.
Chen, C. M.: Optimal points of the stresses for triangular linear element, Numer. Math. J. Chinese Univ. 2 (1980), 12–20 MR 83d:65279.
Chen, C. M.: Superconvergence of finite element solutions and their derivatives, Numer. Math. J. Chinese Univ. 3 (1981), 118–125. MR 82m:65100.
Chen, C. M.: Superconvergence of finite element approximations to nonlinear elliptic problems (Proc. China-France Sympos. on Finite Element Methods, Beijing, 1982), Gordon and Breach, New York, 1983, pp. 622–640. MR 85h:65285.
Chen, C. M.: Superconvergence of finite elements for nonlinear problems, Numer. Math. J. Chinese Univ. 4 (1982), 222–228. MR 83m:65083.
Christiansen, J. and Russell, R. D.: Adaptive mesh selection strategies for solving boundary value problems, SIAM J. Numer. Anal. 15 (1978), 59–80. MR 57#11071.
Christiansen, J. and Russell, R. D.: Error analysis for spline collocation methods with application to knot selection, Math. Comp. 32 (1978), 415–419. MR 58#13736.
Ciarlet, P. G.: The Finite Element Method for Elliptic Problems, North-Holland,Amsterdam, New York, Oxford, 1978. MR 58#25001.
Costabel, M., Stephan, E., and Wendland, W. L.: On the boundary integral equation of the first kind for the bi-Laplacian in a polygonal plane domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Ser. IV, 10 (1983), 197–242. MR 85f:35074.
Cullen, M. J. P.: The use of quadratic finite element methods and irregular grids in the solution of hyperbolic problems, J. Comput. Phys. 45 (1982), 221–245. MR 83f:65152.
Dautov, R. Z.: Superconvergence of difference schemes for the third boundary value problem for quasilinear elliptic equations, Čisl. Metody Meh. Splošn. Sredy 11 (1980), 62–80. MR 84b:65102.
Dautov, R. Z.: Superconvergence of finite-element schemes with numerical integration for quasilinear fourth-order elliptic equations, Differencial'nye Uravnenija 18 (1982), 1172–1181, Differential Equations 18 (1982), 818–824. MR 83k:65080.
Dautov, R. Z. and Lapin, A. V.: Difference schemes of an arbitrary order of accuracy for quasilinear elliptic equations, Izv. Vysš. Učebn. Zaved. Matematika 209 (1979), 24–37. MR 81g:65128.
Dautov, R. Z. and Lapin, A. V.: Investigation of the convergence, in mesh norms, of finite-element-method schemes with numerical integration for fourth-order elliptic equations, Differential Equations 17 (1981), 807–817. MR 83h:65122.
Dautov, R. Z., Lapin, A. V., and Lyashko, A. D.: Some mesh schemes for quasilinear elliptic equations, Ž. Vyčisl. Mat. i Mat. Fiz. 20 (1980), 334–339, U.S.S.R. Computational Math. and Math. Phys. 20 (1980), 62–78. MR 81j:65121.
Descloux, J.: Interior regularity and local convergence of Galerkin finite element approximations for elliptic equations, Topics in Numerical Analysis, II (Proc. Royal Irish Acad. Conf., Dublin 1974), Academic Press, New York, 1975, pp. 24–41. MR 54#1658.
Diaz, J. C.: A collocation-Galerkin method for two-point boundary value problem using continuous piecewise polynomial spaces, SIAM J. Numer. Anal. 14 (1977), 844–858. MR 58#3481.
Dougalis, V. A.: Multistep-Galerkin methods for hyperbolic equations, Math. Comp. 33 (1979), 563–584. MR 81b:65081.
Dougalis, V. A. and Serbin, S. M.: On the superconvergence of Galerkin approximations to second-order hyperbolic equations. SIAM J. Numer. Anal. 17 (1980), 431–446. MR 81j:65106.
Douglas, J. J.: A superconvergence result for the approximate solution of the heat equation by a collocation method, The Mathematical Foundations of the Finite Element method with Applications to Partial Differential Equations Proc. Sympos., Univ. Maryland, Baltimore, 1972). Academic Press, New York, 1972, pp. 475–490. MR 53#7063.
Douglas, J. J.: Improved accuracy through superconvergence in the pressure in the simulation of miscible displacement, Computing Methods in Applied Sciences and Engineering (Proc. Conf., Versailles, 1983), LN in Comp. Sci., North-Holland, 1984, pp. 633–638.
Douglas, J. J.: Superconvergence in the pressure in the simulation of miscible displacement, SIAM J. Numer. Anal. 22 (1985), 962–969. MR 86j:65129.
Douglas, J. J. and Dupont, T.: Superconvergence for Galerkin methods for the two-point boundary problem via local projections, Numer. Math. 21 (1973), 270–278. MR 48#10130.
Douglas, J. J. and Dupont, T.: Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary value problems, Topics in Numerical Analysis, II (Proc. Royal Irish Acad. Conf., Dublin, 1972), Academic Press, London, 1973, pp. 89–93. MR 51#2295.
Douglas, J. J. and Dupont, T.: Galerkin approximations for the two-point boundary problem using continuous, piecewise polynomial spaces. Numer. Math. 22 (1974), 99–109. MR 50#15360.
Douglas, J. J., Dupont, T., and Wahlbin, L., Optimal L ∞ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975), 475–483. MR 51#7298.
Douglas, J. J., Dupont, T., and Wheeler, M. F.: A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems. RAIRO Anal. Numér. 8 (1974), 47–59, MR 50#11811.
Douglas, J. J., Dupont, T. and Wheeler, M. F.: An L ∞ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials, RAIRO Anal. Numér. 8 (1974), 61–66. MR 50#11812.
Douglas, J. J., Ewing, R. E., and Wheeler, M. F.: The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numér. 17 (1983), 17–33.
Douglas, J. J. and Gupta, C. P.: Superconvergence for a mixed finite element method for elastic wave propagation in a plane domain, Numer. Math. 49 (1986), 189–202.
Douglas, J. J. and Milner, F. A.: Interior and superconvergence estimates for mixed methods for second order elliptic problems, RAIRO Modél. Math. Anal. Numér. 19 (1985), 397–428.
Dupont, T.: A unified theory of superconvergence for Galerkin methods for two-point boundary problems, SIAM J. Numer. Anal. 13 (1976), 362–368. MR 53#12021.
Fairweather, G.: Finite Element Galerkin Methods for Differential Equations, Lecture Notes in Pure and Applied Mathematics, Vol. 34, Marcel Dekker, New York, Basel, 1978, MR 58#13781.
Fletcher, C. A. J.: Computational Galerkin Methods, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 85i:65003.
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer Verlag, New York, 1984.
Goldberg, M. A.: A survey of numerical methods for integral equations, Solution Methods for Integral Equations, Math. Concepts and Methods in Sci. and Engrg. Vol. 18. Plenum Press, New York, London, 1979, pp. 1–58. MR 80m:65090.
Goldberg, M. A., Lea, M., and Miel, G.: A superconvergence results for the generalized airfoil equation with application to the flap problem, J. Integral Equations 5 (1983), 175–186. MR 84e:65127.
Graham, I. G.: Galerkin methods for second kind integral equations with singularities, Math. Comp. 39 (1982), 519–533. MR 84d:65090.
Graham, I. G., Joe, S., and Sloan, I. H.: Galerkin versus collocation for integral equations of the second kind, IMA J. Numer. Anal. 5 (1985), 358–369.
de Groen, P. P. N.: A finite element method with a large mesh-width for a stiff two-point boundary value problem, J. Comput. Appl. Math. 7 (1981), 3–15. MR 82d:65059.
Haslinger, J.: Elements finis et l'estimation de l'erreur interieur de la convergence, Comment. Math. Univ. Carolin. 15 (1974), 85–102. MR 49#1799.
El Hatri, M.: Superconvergence of the lumped mass approximation of parabolic equations, C.R. Acad. Bulgare Sci. 36 (1983), 575–578. MR 85h:65187.
El Hatri, M.: Superconvergence of axisymmetrical boundary-value problem, C.R. Acad. Bulgare Sci. 36 (1983), 1499–1502. MR 86e:65153.
El Hatri, M.: Superconvergence in the finite element method for a degenerated boundary value problem, Constr. Theory of Functions (Proc. Conf., Varna, 1984), Bulg. Acad. Sci., Sofia (1984). pp. 328–333.
Hinton, E. and Campbell, J. S.: Local and global smoothing of discontinuous finite element functions using a least squares method, Internat. J. Numer. Methods Engrg. 8 (1974), 461–480. MR 53#14859.
Hinton, E. and Owen, D. R. J.: Finite Element Programming, Academic Press, London, New York, San Francisco, 1977.
Hinton, E., Scott, F. C., and Ricketts, R. E.: Local least squares stress smoothing for parabolic isoparametric elements, Internat. J. Numer. Methods Engrg. 9 (1975), 235–238.
Hlaváček, I. and Křížek, M.: On superconvergent finite element scheme for elliptic systems, Parts I-III, Apl. Mat. 32 (1987), 131–153, 200–213.
De Hoog, F. R. and Weiss, R.: Collocation methods for singular boundary value problems, SIAM J. Numer. Anal. 15 (1978), 198–217. MR 57#8041.
Houstis, E. N.: Application of method of collocation on lines for solving nonlinear hyperbolic problems, Math. Comp. 31 (1977), 443–456. MR 56#1749.
Houstis, E. N.: A collocation method for systems of nonlinear ordinary differential equations, J. Math. Anal. Appl. 62 (1978), 24–37. MR 58#8295.
Hsiao, G. C. and Wendland, W. L.: The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981), 299–315. MR 83j:45019.
Huang, D. and Wu, D.: The superconvergence of the spline finite element solution and its second order derivate for the two-point boundary problem of a fourth order differential equation, J. Zhejiang Univ. 3 (1982), 92–99.
Hulme, B. L.: One-step piecewise polynomial Galerkin methods for initial value problems, Math. Comp. 26 (1972), 415–426. MR 47#9834.
Irons, B. M. and Razzaque, A.: Experience with the patch test for convergence of finite elements, Mathematical Foundations of the Finite Element Method with Applications to PDE, Academic Press, New York, 1972, pp. 557–587. MR 54#11813.
Johnson, C. and Pitkäranta, J.: Analysis of some mixed finite element methods related to reduced integration, Math. Comp. 33 (1982), 375–400. MR 83d:65287.
Kendall, R. P. and Wheeler, M. F.: A Crank-Nicolson H −1-Galerkin procedure for parabolic problems in a single space variable, SIAM J. Numer. Anal. 13 (1976), 861–876. MR 58#13785.
King, J. T. and Serbin, S. M.: Boundary flux estimates for elliptic problems by the perturbed variational method, Computing 16 (1976), 339–347. MR 54#6524.
Korneev, V. G.: Superconvergence of solutions of the finite element method in mesh norms, Ž. Vyčisl. Mat. i Mat. Fiz. 22 (1982), 1133–1148. MR 84b:65114.
Křížek, M. and Neittaanmäki, P.: Superconvergence phenomenon in the finite element method arising from averaging gradients, Numer. Math. 45 (1984), 105–116. MR 86c:65135.
Křížek, M. and Neittaanmäki, P.: Post-processing of a finite element scheme with linear elements, Numerical Techniques in Continuum Mechanics (Proc. 2nd GAMM-Seminar, Kiel, 1986). Notes on Numerical Fluid Mechanics, Vol. 16, 1987, pp. 69–83.
Křížek, M. and Neittaanmäki, P.: On a global superconvergence of the gradient of linear triangular elements, J. Comput. Appl. Math. (1987), (to appear).
Lamp, U., Schleicher, T., Stephan, E., and Wendland, W. L.: Theoretical and experimental asymptotic convergence of the boundary integral method for plane mixed boundary value problem, Boundary Element Methods in Engineering (Proc. Fourth Int. Semin, Southampton, 1982), Springer, Berlin, 1982, pp. 3–17. MR 85e: 3002.
Lapin, A. V.: Mesh schemes of high order of accuracy for some classes of variational inequalities, Sov. J. Numer. Anal. Math. Modelling 2 (1987), 37–55.
Larsen, E. W. and Nelson, P.: Finite-difference approximations and superconvergence for the discrete-ordinate equations in slab geometry, SIAM J. Numer. Anal. 19 (1982), 334–348. MR 83#82050.
Lazarov, R. D., Andreev, A. B., and El Hatri, M.: Superconvergence of the gradient in the finite element method for some elliptic and parabolic problems, Variational-Difference Methods in Mathematical Physics, Part 2 (Proc. Fifth Int. Conf., Moscow, 1983), Viniti, Moscow, 1984, pp. 13–25.
Lesaint, P. and Raviart, P. A.: On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Elements in PDE (Proc. Sympos, Math. Res. Center, Univ. Wisconsin, 1974). Univ. of Wisconsin, Madison, Academic Press, New York, 1974, pp. 89–123. MR 58#31918.
Lesaint, P. and Zlámal, M.: Superconvergence of the gradient of finite element solutions, RAIRO Anal. Numér. 13 (1979), 139–166. MR 80g:65112.
Levine, N.: Stress sampling points for linear triangles in the finite element method, Numerical approximation rep. IMA, Numer. Anal. 5 (1985), 407–427.
Levine, N.: Superconvergent recovery of the gradient from piecewise linear finite element method, Numerical analysis report 6/83, Univ. of Reading, 1983.
Leyk, Z.: Superconvergence in the finite element method, Mat. Stos. 20 (1982), 93–107. MR 85k:65090.
Lin, Q. and Liu, J. Q.: Extrapolation method for Fredholm integral equations with nonsmooth kernels, Numer. Math. 35 (1980), 459–464. MR 81m:65198.
Lin, Q. and Lü, T.: Asymptotic expansions for finite element eigenvalues and finite element solution (Proc. Int. Conf., Bonn, 1983), Math. Schrift. no. 158, Bonn, 1984, pp. 1–10.
Lin, Q. and Lü, T.: Asymptotic expansions for finite element approximation of elliptic problem on polygonal domains, Computing Methods in Applied Sciences and Engineering (Proc. Sixth Int. Conf. Versailles, 1983), LN in Comp. Sci., North-Holland, INRIA, 1984, pp. 317–321.
Lin, Q., Lü, T., and Shen, S.: Asymptotic expansion for finite element approximations, Research Report IMS-11, Inst. Math. Sci. Chengdu Branch of Acad. Sinica, 1983, pp. 1–6. MR 85c:65141.
Lin, Q., Lü, T., and Shen, S.: Maximum norm estimate, extrapolation and optimal point of stresses for the finite element methods on the strongly regular triangulation, J. Comput. Math. 1 (1983), 376–383.
Lin, Q. and Wang, J. P.: Some expansions of the finite element approximation Research Report IMS-15, Academia Sinica, Chengdu Branch, 1984, pp. 1–11. MR 86d:65148.
Lin, Q. and Xu, J. Ch.: Linear finite elements with high accuracy, J. Comput. Math. 3 (1985), 115–133.
Lin, Q. and Zhu, Q. D.: Asymptotic expansion for the derivative of finite elements. J. Comput. Math. 2 (1984), 361–363.
Lindberg, B.: Error estimation and iterative improvement for discretization algorithms, BIT 20 (1980), 486–500. MR 82j:65036.
Locker, J. and Prenter, P. M.: On least squares methods for linear two-point boundary value problems, Functional Analysis Methods in Numerical Analysis (Proc. Spec. Sess., AMS, St. Louis, 1977), LN in Math., Vol. 701, Springer, Berlin, 1979, pp. 149–168. MR 80j:65029.
Locker, J. and Prenter, P. M.: Regularization with differential operators. II: Weak least squares finite element solutions to first kind integral equations, SIAM J. Numer. Anal. 17 (1980), 247–267. MR 83j:65062b.
Louis, A.: Acceleration of convergence for finite element solutions of the Poisson equation, Numer. Math. 33 (1979), 43–53. MR 81i:65091.
Mansfield, L.: On mixed finite element methods for elliptic equations, Comput. Math. Appl. 7 (1981), 59–66. MR 82b:65133.
Marshall, R. S.: An alternative interpretation of superconvergence, Internat. J. Numer. Methods Engrg. 14 (1979), 1707–1709. MR 80m:65079.
Mock, M. S.: Projection methods with different trial and test spaces, Math. Comp. 30 (1976), 400–416. MR 54#11814.
Nakao, M.: Some superconvergence estimates for a collocation H 1-Galerkin method for parabolic problems, Mem. Fac. Sci. Kyushu Univ. Ser. A. 35 (1981), 291–306. MR 83a:65111.
Nakao, M.: Interior estimates and superconvergence for H −1-Galerkin method to elliptic equations, Bull. Kyushu Inst. Tech. Math. Natur. Sci. 30 (1983), 19–30. MR 85i:65154.
Nakao, M.: Superconvergence estimates at Jacobi points of the collocation-Galerkin method for two-point boundary value problems J. Inform. Process. 7 (1984), 31–34. MR 85m:65079.
Nakao, M.: Some superconvergence estimates for a Galerkin method for elliptic problems, Bull. Kyushu Inst. Tech. Math. Natur. Sci. 31 (1984), 49–58. MR 86e:65159.
Nakao, M.: Some superconvergence of Galerkin approximations for parabolic and hyperbolic problems in one space dimension, Bull. Kyushu Inst. Tech. Math. Natur. Sci. 32 (1985), 1–14. MR 86j:65123.
Neittaanmäki, P. and K<rí<zek, M.: Superconvergence of the finite element schemes arising from the use of averaged gradients, Accuracy Estimates and Adaptive Refinements in Finite Element Computations (Proc. Int. Conf., Lisbon, 1984), pp. 169–178.
Neittaanmäki, P. and Saranen, J.: A modified least squares FE-method for ideal fluid flow problems, J. Comput. Appl. Math. 8 (1982), 165–169.
Neta, B. and Victory, H. D.: A new fourth-order finite-difference method for solving discreteordinates slab transport equations, SIAM J. Numer. Aral. 20 (1983), 94–105. MR 84c:65082.
Nitsche, J. A.: L ∞-error analysis for finite elements, Mathematics of Finite Elements and Applications, III (Proc. Third MAFELAP Conf., Brunel Univ., Uxbridge, 1978), Academic Press, London, 1979, pp. 173–186. MR 81f:65084.
Nitsche, J. A. and Schatz, A. H.: Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 51#9525.
Oganesjan, L. A., Rivkind, V. J., and Ruhovec, L. A.: Variation-difference methods for the solution of elliptic equations, Part 1 Proc. Sem., Issue 5, Vilnius, 1973), Inst. of Phys. and Math., Vilnius, 1973, pp. 3–389, MR 57#18146.
Oganesjan, L. A. and Ruhovec, L. A.: An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary, <Z. Vyčisl. Mat. i Mat. Fiz. 9 (1969), 1102–1120. MR 45#4665.
Oganesjan, L. A. and Ruhovec, L. A.: Variational-Difference Methods for the Solution of Elliptic Equations, Izd. Akad. Nauk Armjanskoi SSR, Jerevan, 1979. MR 82m:65105.
Prenter, P. M. and Locker, J.: Optimal L 2 and L ∞ error estimates for continuous and discrete least squares methods for boundary value problems, SIAM J. Numer. Anal. 15 (1978), 1151–1160. MR 80d:65097.
Pereyra, W. and Sewell, E. G.: Mesh selection for discrete solution of boundary value problems in ordinary differential equations, Numer. Math. 23 (1975), 261–268. MR 57#4527.
Rachford, H. H. and Wheeler, M. F.: An H −1-Galerkin procedure for the two-point boundary value problem, Mathematical Aspects of Finite Elements in Partial Diff. Equations, Academic Press, New York, 1974, pp. 353–382. MR 50#1525.
Rannacher, R. and Scott, R.: Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), 437–445.
Reddien, G. W.: Collocation at Gauss points as a discretization in optimal control, SIAM J. Control Optim. 17 (1979), 298–306. MR 80b:93171.
Reddien, G. W.: Codes for boundary-value problems in ordinary differential equations (Proc. Conf. Houston, 1978.) LN in Comp. Sci., Vol. 76, Springer, Berlin, 1979, pp. 206–227.
Reddien, G. W.: Projection methods for two-point boundary value problems, SIAM Rev. 22 (1980), 156–171. MR 81e:65049.
Regińska, T.: Superconvergence of external approximation of eigenvalues of ordinary differential operators, IMA J. Numer. Anal. 6 (1986), 309–323.
Richter, G. R.: Superconvergence of piecewise polynomial Galerkin approximations, for Fredholm integral equations of the second kind, Numer. Math. 31 (1978/79), 63–70. MR 80a:65273.
Saranen, J. and Wendland, W. L.: On the asymptotic convergence of collocation methods with spline function of even degree, Math. Comp. 45 (1983), 91–108.
Schäfer, E.: Fehlerabschätzungen für Eigenwertnäherungen nach der Ersatzkernmethode bei Integralgleichungen, Numer. Math. 32 (1979), 281–290. MR 80e:65129.
Schatz, A. H. and Wahlbin, L. B.: Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), 414–442. MR 55#4748.
Von Seggern, R.: Die Methode der finiten Elemente be der numerischen Behandlung linear Integrodifferentialgleichungen, Kernforschungsanlage Julich, Zentralinstitut für Angewandte Mathematik, Julich, 1979. MR 83e:65215.
Von Seggern, R.: Ein Superkonvergenzresultat bei Anwendung der Methode der finiten Elemente auf lineare Integrodifferentialgleichungen, Z. Angew. Math. Mech. 61 (1981), 320–321. MR 84e:45009.
Sloan, I. H.: Iterated Galerkin method for eigenvalue problems, SIAM J. Numer. Anal. 13 (1976), 753–760. MR 55#1727.
Sloan, I. H.: A review of numerical methods for Fredholm integral equations of the second kind, Application and Numerical Solution of Integral Equations (Proc. Sem., Australian Nat. Univ., Canberra, 1978), Sijthoff & Noordhoff, Alphen-aan-den-Rijn, 1980, pp. 51–74. MR 81h:65133.
Sloan, I. H.: Superconvergence and the Galerkin method for integral equations of the second kind, Treatment of Integral Equations by Numerical Methods (Proc. Symp., Durham, 1982). Academic Press, New York, London, 1982, pp. 197–207. MR 85e:65005.
Sloan, I. H.: Four variants of the Galerkin method for integral equations of the second kind, IMA J. Numer. Anal. 4 (1984), 9–17. MR 85h:65277.
Sloan, I. H. and Thomée, V.: Superconvergence of the Galerkin iterates for integral equations of the second kind. J. Integral Equations 9 (1985), 1–23. MR 86j:65184.
Spence, A. and Thomas, K. S.: On superconvergence properties of Galerkin's method for compact operator equations, IMA J. Numer. Anal. 3 (1983), 253–271. MR 85c:65074.
Stephan, E. and Wendland, W. L.: An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems. Applicable Anal. 18 (1984), 183–220.
Strang, G. and Fix, G.: An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, New Jersey, 1973. MR 56#1747.
Thomée, V.: Spline approximation and difference schemes for the heat equation. Math. Foundations of the Finite Element Method with Applications to PDE (Proc. Sympos., Univ. of Maryland, 1972). Academic Press, New York, London, 1972, pp. 711–746. MR 49#11824.
Thomée, V.: Convergence estimates for semi-discrete Galerkin methods for initial-value problems, Numerische, insbesondere approximations-theoretische Behandlung von Funktionalgleichungen. LN in Math., Vol. 333, Springer Verlag, New York, 1973, pp. 243–262. MR 56#17147.
Thomée, V.: Some error estimates in Galerkin methods for parabolic equations, Mathematical Aspects of Finite Element Methods (Proc. Conf., Rome, 1975), LN in Math., Vol. 606, Springer-Verlag, Berlin, Heidelberg, New York, 1977, pp. 353–362. MR 58#31928.
Thomée, V.: High order local approximations to derivatives in the finite element method, Math. Comp. 31 (1977), pp. 652–660. MR 55#11572.
Thomée, V.: Galerkin-finite element method for parabolic equations, Bull. Iranian Math. Soc. 10 (1978), 3–17. MR 81b:65107.
Thomée, V.: Galerkin-finite element methods for parabolic equations (Proc. Internat. Congress of Mathematicians, Helsinki, 1978). Acad. Sci. Fennica, Helsinki, 1980, pp. 943–952. MR 81i:65100.
Thomée, V.: Negative norm estimates and superconvergence in Galerkin methods for parabolic problems, Math. Comp. 34 (1980), 93–113. MR 81a:65092.
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, LN in Math., Vol. 1054, Springer-Verlag, Berlin, 1984. MR 86k:65006.
Thomée, V. and Wendroff, B.: Convergence estimates for Galerkin method for variable coefficient initial value problems. SIAM J. Numer. Anal. 11 (1974), 1059–1068. MR 51#7309.
Thomée, V. and Westergren, B.: Elliptic difference equations and interior regularity, Numer. Math. 11 (1968), 196–210. MR 36#7347.
Vacek, J.: Dual variational principle for an elliptic partial differential equation, Apl. Mat. 21 (1976), 5–27. MR 54#716.
Veryard, O. A.: Problems associated with the convergence of isoparametric and miscoparametric finite elements, MSc thesis, Univ. of Wales, 1971.
Volk, W.: Globale Superconvergenz bei der Lösung von Differentialaufgaben, Numer. Math. 48 (1986), 617–625.
Wendland, W. L.: Boundary element methods and their asymptotic convergence, Theoretical Acoustics and Numerical Techniques, CISM Courses No. 277, Springer, Wien, New York, 1983, pp. 135–216.
Wendland, W. L., Stephan, E., and Hsiao, G. C.: On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Methods Appl. Sci. 1 (1979), 265–321. MR 82e:31003.
Westergren, B.: Interior estimates for elliptic systems of difference equations, PhD thesis, Univ. of Göteborg, 1982.
Wheeler, M. F.: An optimal L ∞ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal. 10 (1973), 914–917. MR 49#8399.
Wheeler, M. F.: Galerkin procedure for estimating the flux for two-point boundary value problems, SIAM J. Numer. Anal. 11 (1974), 764–768. MR 52#4644.
Wheeler, M. F.: A C 0-collocation-finite element method for two-point boundary value problems and one space dimensional parabolic problems, SIAM J. Numer. Anal. 14 (1977), 71–90. MR 56#13667.
Wielgosz, C.: Exact results given by finite element methods in mechanics, J. Méc. Appl. 1 (1982), 323–329.
Winther, R.: Some superlinear convergent results for the conjugate gradient method, SIAM J. Numer. Anal. 17 (1980), 14–17. MR 81k: 65060.
Winther, R.: A stable finite element method for initial-boundary value problems for first-order hyperbolic systems, Math. Comp. 36 (1981), 65–86. MR 81m:65181.
Winther, R.: A finite element method for a version of the Boussinesq equation, SIAM J. Numer. Anal. 19 (1982), 561–570. MR 83f:65184.
Zhu, Q. D.: A superconvergence result for the finite element method, Numer. Math. J. Chinese Univ. 3 (1981), 50–55. MR 82f:65121.
Zhu, Q. D.: Natural inner superconvergence for the finite element method. Proc. China-France Sympos. on Finite Element Methods, Beijing, 1982), Gordon and Breach, New York, 1983, pp. 935–960. MR 85h:65253.
Zhu, Q. D.: Uniform superconvergence estimates of derivates for the finite element method, Numer. Math. J. Chinese Univ. 5 (1983), 311–318. MR 85k:65096.
Zienkiewicz, O. C. and Cheung, Y. K., The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London, 1967. MR 47#4518.
Zlámal, M.: Some superconvergence results in the finite element method, Mathematical Aspects of Finite Element Methods (Proc. Conf., Rome, 1975), LN in Math. Vol. 606, Springer-Verlag, Berlin, Heidelberg, New York, 1977, pp. 353–362. MR 58#8365.
Zlámal, M.: Superconvergence and reduced integration in the finite element method, Math. Comp. 32 (1978), 663–685. MR 58#13794.
Zlámal, M.: Superconvergence of gradients in the finite element method, Variational-Difference Methods in Mathematical Physics (Proc. Third. Conf., Novosibirsk, 1977), Akad. Nauk. SSSR Sibirsk. Otdel., Vychisl. Centr, Novosibirsk, 1978, pp. 15–22. MR 82e:65116.
Zlámal, M.: Superconvergence of the gradient of finite element solutions (Wiss. Z. Hochschule Architektur Bauwesen Weimar, 1979). Weimar, 1979, pp. 375–380.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Křížek, M., Neittaanmäki, P. On superconvergence techniques. Acta Appl Math 9, 175–198 (1987). https://doi.org/10.1007/BF00047538
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00047538