Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Analysis of optimal finite-element meshes in $\textbf {R}^{1}$
HTML articles powered by AMS MathViewer

by I. Babuška and W. C. Rheinboldt PDF
Math. Comp. 33 (1979), 435-463 Request permission

Abstract:

A theory of a posteriori estimates for the finite-element method was developed earlier by the authors. Based on this theory, for a two-point boundary value problem the existence of a unique optimal mesh distribution is proved and its properties analyzed. This mesh is characterized in terms of certain, easily computable local error indicators which in turn allow for a simple adaptive construction of the mesh and also permit the computation of a very effective a posteriori error bound. While the error estimates are asymptotic in nature, numerical experiments show the results to be excellent already for 10
References
  • Naum I. Akhiezer, The calculus of variations, A Blaisdell Book in the Pure and Applied Sciences, Blaisdell Publishing Co. (a division of Random House), New York-London, 1962. Translated from the Russian by Aline H. Frink. MR 0142019
    • I. BABUŠKA, "The self-adaptive approach in the finite element method," The Mathematics of Finite Elements and Applications II, MAFELAP 1975 (J. R. Whiteman, Ed.), Academic Press, London, 1976, pp. 125-142. I. BABUŠKA & W. RHEINBOLDT, A-Posteriori Error Estimates for the Finite Element Method, Computer Science Technical Report TR-581, University of Maryland, 1977; Internat. J. Numer. Methods Engrg., v. 12, 1978, pp. 1597-1615.
  • I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 483395, DOI 10.1137/0715049
  • Werner C. Rheinboldt, Numerical continuation methods for finite element applications, Formulations and computational algorithms in finite element analysis (U.S.-Germany Sympos., Mass. Inst. Tech., Cambridge, Mass., 1976) M.I.T. Press, Cambridge, Mass., 1977, pp. 599–631. MR 0474782
    • I. BABUSKA, W. RHEINBOLDT & C. MESZTENYI, Self-Adaptive Refinement in the Finite Element Method, Computer Science Technical Report TR-375, University of Maryland, 1975.
  • Hermann G. Burchard, Splines (with optimal knots) are better, Applicable Anal. 3 (1973/74), 309–319. MR 399708, DOI 10.1080/00036817408839073
  • H. G. Burchard and D. F. Hale, Piecewise polynomial approximation on optimal meshes, J. Approximation Theory 14 (1975), no. 2, 128–147. MR 374761, DOI 10.1016/0021-9045(75)90084-2
  • W. E. Carroll and R. M. Barker, A theorem for optimum finite-element idealizations, Internat. J. Solids Structures 9 (1973), 883–895 (English, with Russian summary). MR 337119, DOI 10.1016/0020-7683(73)90011-5
  • Carl de Boor, Good approximation by splines with variable knots, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) Internat. Ser. Numer. Math., Vol. 21, Birkhäuser, Basel, 1973, pp. 57–72. MR 0403169
    • C. deBOOR, Good Approximations with Variable Knots. II, Lecture Notes in Math., Vol. 363, Springer-Verlag, Berlin, 1973, pp. 12-20. C. deBOOR & J. R. RICE, An Adaptive Algorithm for Multivariate Approximation Giving Optimal Convergence Rates, Mathematics Research Center, MRC Technical Summary Report 1773, University of Wisconsin, 1977.
  • Carl de Boor and Blâir Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582–606. MR 373328, DOI 10.1137/0710052
  • V. E. Denny and R. B. Landis, A new method for solving two-point boundary value problems using optimal node distribution, J. Comput. Phys. 9 (1972), 120–137. MR 295583, DOI 10.1016/0021-9991(72)90039-3
  • T. E. Hull, Numerical solutions of initial value problems for ordinary differential equations, Numerical solutions of boundary value problems for ordinary differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1974) Academic Press, New York, 1975, pp. 3–26. MR 0403226
  • H. B. Keller, Numerical solution of boundary value problems for ordinary differential equations: survey and some recent results on difference methods, Numerical solutions of boundary value problems for ordinary differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1974) Academic Press, New York, 1975, pp. 27–88. MR 0451742
  • M. Lentini and V. Pereyra, Boundary problem solvers for first order systems based on deferred corrections, Numerical solutions of boundary value problems for ordinary differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1974) Academic Press, New York, 1975, pp. 293–315. MR 0488787
    • R. J. MELOSH & D. E. KILLIAN, Finite Element Analysis to Attain a Prespecified Accuracy, Presented at 2nd National Symposium on Computerized Structural Analysis, George Washington University, Washington, D.C., 1976. (Preprint.) R. J. MELOSH & P. V. MARCAL, "An energy basis for mesh refinement of structural continua," Internat. J. Numer. Methods Engrg., v. 11, 1977, pp. 1083-1091.
  • Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI 10.1090/S0025-5718-1974-0373325-9
  • G. M. Phillips, Error estimates for best polynomial approximations, Approximation Theory (Proc. Sympos., Lancaster, 1969) Academic Press, London, 1970, pp. 1–6. MR 0277970
  • Carl E. Pearson, On a differential equation of boundary layer type, J. Math. and Phys. 47 (1968), 134–154. MR 228189, DOI 10.1002/sapm1968471134
    • G. SEWELL, "An adaptive computer program for the solution of ${\operatorname {Div}}(p(x,y)\operatorname {grad}\,u) = f(x,y,u)$ on a polygonal region," The Mathematics of Finite Elements and Applications II, MAFELAP 1975 (J. Whiteman, Ed.), Academic Press, London, 1976, pp. 125-144.
  • Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
    • D. J. TURCKE, "Further developments in grid selection procedures in the finite element method," Proc. Fifth Canadian Congress Appl. Mech., University of New Brunswick, Frederickton, N.B., Canada, 1975. D. J. TURCKE & G. M. McNEICE, "A variational approach to grid optimization in the finite element method," Conf. on Variational Methods in Engineering, Southhampton University, England, 1972. D. J. TURCKE & G. M. McNEICE, "Guidelines for selecting finite element grids based on an optimization study," Computers and Structures, v. 4, 1973, pp. 499-519.
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 65N30
  • Retrieve articles in all journals with MSC: 65N30
Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Math. Comp. 33 (1979), 435-463
  • MSC: Primary 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-1979-0521270-2
  • MathSciNet review: 521270