Analysis of optimal finiteelement meshes in
Authors:
I. Babuška and W. C. Rheinboldt
Journal:
Math. Comp. 33 (1979), 435463
MSC:
Primary 65N30
MathSciNet review:
521270
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Abstract: A theory of a posteriori estimates for the finiteelement method was developed earlier by the authors. Based on this theory, for a twopoint 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
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 [1]
 N. I. AKHIEZER, The Calculus of Variations, Translated from the Russian by A. H. Frink, Blaisdell, New York, 1962. MR 0142019 (25:5414)
 [2]
 I. BABUŠKA, "The selfadaptive 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. 125142.
 [3]
 I. BABUŠKA & W. RHEINBOLDT, APosteriori Error Estimates for the Finite Element Method, Computer Science Technical Report TR581, University of Maryland, 1977; Internat. J. Numer. Methods Engrg., v. 12, 1978, pp. 15971615.
 [4]
 I. BABUŠKA & W. RHEINBOLDT, Error Estimates for Adaptive Finite Element Computations, Computer Science Technical Report BN854, University of Maryland, 1911; SIAM J. Numer. Anal., v. 15, 1978, pp. 736754. MR 0483395 (58:3400)
 [5]
 I. BABUŠKA & W. RHEINBOLDT, "Computational aspects of finite element analysis," Mathematical SoftwareIII (J. R. Rice, Ed.), Academic Press, New York, 1977, pp. 223253. MR 0474782 (57:14415)
 [6]
 I. BABUSKA, W. RHEINBOLDT & C. MESZTENYI, SelfAdaptive Refinement in the Finite Element Method, Computer Science Technical Report TR375, University of Maryland, 1975.
 [7]
 H. G. BURCHARD, "Splines (with optimal knots) are better," J. Appl. Anal., v. 3, 1974, pp. 309319. MR 0399708 (53:3551)
 [8]
 H. G. BURCHARD & D. F. HALE, "Piecewise polynomial approximations on optimal meshes," J. Approximation Theory, v. 14, 1975, pp. 128147. MR 0374761 (51:10957)
 [9]
 W. E. CARROLL & R. M. BARKER, "A theorem for optimum finite element idealization," Inst. J. Solids Structures, v. 9, 1973, pp. 883895. MR 0337119 (49:1891)
 [10]
 C. deBOOR, "Good approximation by splines with variable knots," Spline Functions and Approximation Theory (A. Meir, A. Sharma, Eds.), ISNM Vol. 21, Birkhäuser Verlag, Basel, 1973, pp. 5772. MR 0403169 (53:6982)
 [11]
 C. deBOOR, Good Approximations with Variable Knots. II, Lecture Notes in Math., Vol. 363, SpringerVerlag, Berlin, 1973, pp. 1220.
 [12]
 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.
 [13]
 C. deBOOR & B. SWARTZ, "Collocation at Gaussian points," SIAM J. Numer. Anal., v. 10, 1973. MR 0373328 (51:9528)
 [14]
 V. E. DENNY & R. B. LANDIS, "A new method for solving twopoint boundary value problems using optimal node distributions," J. Computational Phys., v. 9, 1972, pp. 120137. MR 0295583 (45:4649)
 [15]
 T. E. HULL, "Numerical solution of initial value problems for ordinary differential equations," Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations (A. K. Aziz, Ed.), Academic Press, New York, 1975, pp. 326. MR 0403226 (53:7038)
 [16]
 H. B. KELLER, "Numerical solution of boundary value problems for ordinary differential equations: Survey and some recent results on difference methods," Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (A. K. Aziz, Ed.), Academic Press, New York, 1975, pp. 2788. MR 0451742 (56:10024)
 [17]
 M. LENTINI & V. PEREYRA, "Boundary problem solvers for firstorder systems based on deferred corrections," Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (A. K. Aziz, Ed.), Academic Press, New York, 1975, pp. 293316. MR 0488787 (58:8297)
 [18]
 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.)
 [19]
 R. J. MELOSH & P. V. MARCAL, "An energy basis for mesh refinement of structural continua," Internat. J. Numer. Methods Engrg., v. 11, 1977, pp. 10831091.
 [20]
 J. A. NITSCHE & A. H. SCHATZ, "Interior estimates for RitzGalerkin methods," Math. Comp., v. 28, 1974, pp. 937948. MR 0373325 (51:9525)
 [21]
 G. M. PHILLIPS, "Error estimates for best approximation," Approximation Theory (A. Talbot, Ed.), Academic Press, London, 1970, pp. 16. MR 0277970 (43:3703)
 [22]
 C. E. PEARSON, "On a differential equation of boundary layer type," J. Math. and Phys., v. 47, 1968, pp. 134154. MR 0228189 (37:3773)
 [23]
 G. SEWELL, "An adaptive computer program for the solution of on a polygonal region," The Mathematics of Finite Elements and Applications II, MAFELAP 1975 (J. Whiteman, Ed.), Academic Press, London, 1976, pp. 125144.
 [24]
 G. STRANG & G. J. FIX, An Analysis of the Finite Element Method, PrenticeHall, Englewood Cliffs, N.J., 1973. MR 0443377 (56:1747)
 [25]
 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.
 [26]
 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.
 [27]
 D. J. TURCKE & G. M. McNEICE, "Guidelines for selecting finite element grids based on an optimization study," Computers and Structures, v. 4, 1973, pp. 499519.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905212702
PII:
S 00255718(1979)05212702
Article copyright:
© Copyright 1979 American Mathematical Society
