Analysis of optimal finite-element meshes in

Authors:
I. Babuška and W. C. Rheinboldt

Journal:
Math. Comp. **33** (1979), 435-463

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521270-2

MathSciNet review:
521270

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

**[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 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.**[3]**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.**[4]**I. BABUŠKA & W. RHEINBOLDT,*Error Estimates for Adaptive Finite Element Computations*, Computer Science Technical Report BN-854, University of Maryland, 1911;*SIAM J. Numer. Anal.*, v. 15, 1978, pp. 736-754. MR**0483395 (58:3400)****[5]**I. BABUŠKA & W. RHEINBOLDT, "Computational aspects of finite element analysis,"*Mathematical Software*-III (J. R. Rice, Ed.), Academic Press, New York, 1977, pp. 223-253. MR**0474782 (57:14415)****[6]**I. BABUSKA, W. RHEINBOLDT & C. MESZTENYI,*Self-Adaptive Refinement in the Finite Element Method*, Computer Science Technical Report TR-375, University of Maryland, 1975.**[7]**H. G. BURCHARD, "Splines (with optimal knots) are better,"*J. Appl. Anal.*, v. 3, 1974, pp. 309-319. MR**0399708 (53:3551)****[8]**H. G. BURCHARD & D. F. HALE, "Piecewise polynomial approximations on optimal meshes,"*J. Approximation Theory*, v. 14, 1975, pp. 128-147. 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. 883-895. 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. 57-72. MR**0403169 (53:6982)****[11]**C. deBOOR,*Good Approximations with Variable Knots*. II, Lecture Notes in Math., Vol. 363, Springer-Verlag, Berlin, 1973, pp. 12-20.**[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 two-point boundary value problems using optimal node distributions,"*J. Computational Phys.*, v. 9, 1972, pp. 120-137. 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. 3-26. 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. 27-88. MR**0451742 (56:10024)****[17]**M. LENTINI & V. PEREYRA, "Boundary problem solvers for first-order 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. 293-316. 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. 1083-1091.**[20]**J. A. NITSCHE & A. H. SCHATZ, "Interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 28, 1974, pp. 937-948. MR**0373325 (51:9525)****[21]**G. M. PHILLIPS, "Error estimates for best approximation,"*Approximation Theory*(A. Talbot, Ed.), Academic Press, London, 1970, pp. 1-6. MR**0277970 (43:3703)****[22]**C. E. PEARSON, "On a differential equation of boundary layer type,"*J. Math. and Phys.*, v. 47, 1968, pp. 134-154. 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. 125-144.**[24]**G. STRANG & G. J. FIX,*An Analysis of the Finite Element Method*, Prentice-Hall, 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. 499-519.

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30

Retrieve articles in all journals with MSC: 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521270-2

Article copyright:
© Copyright 1979
American Mathematical Society