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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Some a posteriori error estimators for elliptic partial differential equations


Authors: R. E. Bank and A. Weiser
Journal: Math. Comp. 44 (1985), 283-301
MSC: Primary 65N30; Secondary 65N15
MathSciNet review: 777265
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Abstract: We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations. The estimators are based on solving local Neumann problems in each element. The estimators differ in how they enforce consistency of the Neumann problems. We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error. We present numerical results in which one of the error estimators appears to converge to the norm of the true error.


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  • [1] Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0178246 (31 #2504)
  • [2] I. Babuška, Private Communication.
  • [3] I. Babuška & M. Luskin, "An adaptive time discretization procedure for parabolic problems," in Proc. Fourth IMACS Internat. Sympos. on Computer Methods for Partial Differential Equations, Lehigh University, Bethlehem, Pennsylvania, 1981, pp. 5-8.
  • [4] I. Babuška & A. Miller, A Posteriori Error Estimates and Adaptive Techniques for the Finite Element Method, Technical Report BN-968, Institute for Physical Science and Technology, University of Maryland, 1981.
  • [5] Ivo Babuška and Werner C. Rheinboldt, A posteriori error analysis of finite element solutions for one-dimensional problems, SIAM J. Numer. Anal. 18 (1981), no. 3, 565–589. MR 615532 (82j:65082), http://dx.doi.org/10.1137/0718036
  • [6] I. Babuška & W. C. Rheinboldt, "A posteriori error estimates for the finite element method," Internat. J. Numer. Methods Engrg., v. 12, 1978, pp. 1597-1615.
  • [7] I. Babuška and W. C. Rheinboldt, Analysis of optimal finite-element meshes in 𝑅¹, Math. Comp. 33 (1979), no. 146, 435–463. MR 521270 (80b:65134), http://dx.doi.org/10.1090/S0025-5718-1979-0521270-2
  • [8] I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 0483395 (58 #3400)
  • [9] I. Babuška & W. C. Rheinboldt, "On the reliability and optimality of the finite element method," Comput. & Structures, v. 10, 1979, 87-94.
  • [10] Ivo Babuška and Werner C. Rheinboldt, Reliable error estimation and mesh adaptation for the finite element method, Computational methods in nonlinear mechanics (Proc. Second Internat. Conf., Univ. Texas, Austin, Tex., 1979) North-Holland, Amsterdam-New York, 1980, pp. 67–108. MR 576902 (82f:65109)
  • [11] R. E. Bank, PLTMG Users' Guide, June, 1981 version, Technical Report, Department of Mathematics, University of California, San Diego, 1982.
  • [12] R. E. Bank & T. F. Dupont, Analysis of a Two-Level Scheme for Solving Finite Element Equations, Technical Report, University of Texas Center for Numerical Analysis, 1980.
  • [13] R. E. Bank & A. H. Sherman, "A multi-level iterative method for solving finite element equations," in Proc. Fifth Symposium on Reservoir Simulation, Society of Petroleum Engineers of AIME, Dallas, 1979, pp. 117-126.
  • [14] Peter Percell and Mary Fanett Wheeler, A local residual finite element procedure for elliptic equations, SIAM J. Numer. Anal. 15 (1978), no. 4, 705–714. MR 0495021 (58 #13789)
  • [15] Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR 0443377 (56 #1747)
  • [16] A. Weiser, Local-Mesh, Local-Order, Adaptive Finite Element Methods With A Posteriori Error Estimators for Elliptic Partial Differential Equations, Technical Report 213, Computer Science Department, Yale University, 1981.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1985-0777265-X
PII: S 0025-5718(1985)0777265-X
Article copyright: © Copyright 1985 American Mathematical Society