Some a posteriori error estimators for elliptic partial differential equations
Authors:
R. E. Bank and A. Weiser
Journal:
Math. Comp. 44 (1985), 283301
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.
 [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.TorontoLondon, 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. 58.
 [4]
I. Babuška & A. Miller, A Posteriori Error Estimates and Adaptive Techniques for the Finite Element Method, Technical Report BN968, 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 onedimensional 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. 15971615.
 [7]
I.
Babuška and W.
C. Rheinboldt, Analysis of optimal finiteelement
meshes in 𝑅¹, Math. Comp.
33 (1979), no. 146, 435–463. MR 521270
(80b:65134), http://dx.doi.org/10.1090/S00255718197905212702
 [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, 8794.
 [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)
NorthHolland, AmsterdamNew 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 TwoLevel Scheme for Solving Finite Element Equations, Technical Report, University of Texas Center for Numerical Analysis, 1980.
 [13]
R. E. Bank & A. H. Sherman, "A multilevel iterative method for solving finite element equations," in Proc. Fifth Symposium on Reservoir Simulation, Society of Petroleum Engineers of AIME, Dallas, 1979, pp. 117126.
 [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,
PrenticeHall, Inc., Englewood Cliffs, N. J., 1973. PrenticeHall Series in
Automatic Computation. MR 0443377
(56 #1747)
 [16]
A. Weiser, LocalMesh, LocalOrder, Adaptive Finite Element Methods With A Posteriori Error Estimators for Elliptic Partial Differential Equations, Technical Report 213, Computer Science Department, Yale University, 1981.
 [1]
 S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, New York, 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. 58.
 [4]
 I. Babuška & A. Miller, A Posteriori Error Estimates and Adaptive Techniques for the Finite Element Method, Technical Report BN968, Institute for Physical Science and Technology, University of Maryland, 1981.
 [5]
 I. Babuška & W. C. Rheinboldt, "A posteriori error analysis of finite element solutions for onedimensional problems," SIAM J. Numer. Anal., v. 18, 1981, pp. 565589. MR 615532 (82j:65082)
 [6]
 I. Babuška & W. C. Rheinboldt, "A posteriori error estimates for the finite element method," Internat. J. Numer. Methods Engrg., v. 12, 1978, pp. 15971615.
 [7]
 I. Babuška & W. C. Rheinboldt, "Analysis of optimal finiteelement meshes in ," Math. Comp., v. 33, 1979, pp. 435463. MR 521270 (80b:65134)
 [8]
 I. Babuška & W. C. Rheinboldt, "Error estimates for adaptive finite element computations," SIAM J. Numer. Anal., v. 15, 1978, 736754. 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, 8794.
 [10]
 I. Babuška & W. C. Rheinboldt, "Reliable error estimation and mesh adaptation for the finite element method," in Computational Methods in Nonlinear Mechanics, NorthHolland, New York, 1980, pp. 67108. 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 TwoLevel Scheme for Solving Finite Element Equations, Technical Report, University of Texas Center for Numerical Analysis, 1980.
 [13]
 R. E. Bank & A. H. Sherman, "A multilevel iterative method for solving finite element equations," in Proc. Fifth Symposium on Reservoir Simulation, Society of Petroleum Engineers of AIME, Dallas, 1979, pp. 117126.
 [14]
 P. Percell & M. F. Wheeler, "A local residual finite element procedure for elliptic equations," SIAM J. Numer. Anal., v. 15; 1978, pp. 705714. MR 0495021 (58:13789)
 [15]
 G. Strang & G. J. Fix, An Analysis of the Finite Element Method, PrenticeHall, Englewood Cliffs, N. J., 1973. MR 0443377 (56:1747)
 [16]
 A. Weiser, LocalMesh, LocalOrder, 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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DOI:
http://dx.doi.org/10.1090/S0025571819850777265X
PII:
S 00255718(1985)0777265X
Article copyright:
© Copyright 1985
American Mathematical Society
