A posteriori error estimation techniques for non-linear elliptic and parabolic pdes
Keywords:
a posteriori error estimates, non-linear problems, elliptic pdes, parabolic pdesAbstract
We give a brief overview of a posteriori error estimation techniques for nonlinear elliptic and parabolic pdes and point out some related questions which are not yet satisfactorily settled.
Downloads
References
[ADJ 88] ADJERID S., FLAHERTY J. E., «A local refinement finite element
method for two dimensional parabolic problems», SIAM J. Sci. Stat. Comput.,
vol. 9, 1988, p. 792- 811.
[BAB 78] BABUSKA I., RHEINBOLDT W. C., «Error estimates for adaptive finite
element computations», SIAM J. Numer. Anal., vol. 15, 1978, p. 736- 754.
[BAN 85] BANK R. E., WEISER A., «Some a posteriori error estimators for
elliptic partial differential equations», Math. Comput., vol. 44, 1985, p. 283
- 301.
[BEC 95] BECKER R., RANNACHER R., «Weighted a posteriori error control in
FE methods», Proc. ENUMATH-95, Paris, September 1995
[BER 00] BERNARDI CH., VERFURTH R., «Adaptive finite element methods for
elliptic equations with non-smooth coefficients», Numer. Math., 2000, (to
appear)
[BIE 82aj BIETERMAN M., BABUSKA I., «The finite element method for parabolic
equations I. A posteriori error estimation», Numer. Math., vol. 40, 1982,
p. 339- 371.
[BIE 82bj BIETERMAN M., BABUSKA I., «The finite element method for parabolic
equations II. A posteriori error estimation and adaptive approach»,
Numer. Math., vol. 40, 1982, p. 373 - 406.
[BOR 90] BORNEMANN F., «An adaptive multilevel approach to parabolic equations
I. General theory and 1D implementation», IMPACT of Camp. in Sci.
and Engrg., vol. 2, 1990, 279- 317.
[BOR 91] BoRNEMANN F., «An adaptive multilevel approach to parabolic equations
II. Variable order time discretization based on a multiplicative error
correction», IMPACT of Camp. in Sci. and Engrg., vol. 3, 1991, p. 93- 122.
[BOR 92] BoRNEMANN F., «An adaptive multilevel approach to parabolic equations
III. 2D error estimation and multilevel preconditioning», IMPACT of
Camp. in Sci. and Engrg., vol. 4, 1992, p. 1 - 45.
[ERI 91] ERIKSSON K., JOHNSON C., «Adaptive finite element methods for parabolic
problems I. A linear model problem», SIAM J. Numer. Anal., vol.
, 1991, p. 43 - 77.
[ERI 95aj ERIKSSON K., JOHNSON C., «Adaptive finite element methods for
parabolic problems IV. Nonlinear problems», SIAM J. Numer. Anal., vol.
, 1995, p. 1724-1749.
[ERI 95bj ERIKSSON K., JOHNSON C., «Adaptive finite element methods for
parabolic problems V. Long-time integration», SIAM J. Numer. Anal., vol.
, 1995, p. 1750 - 1763.
[ERI 85] ERIKSSON K., JOHNSON C., THOMEE V., «Time discretization of parabolic
problems by the discontinuous Galerkin method», Modelisation Mathematique
et Analyse Numerique, vol. 19, 1985, p. 611 - 643.
[KUN 98] KuNERT G., A posteriori error estimation for anisotropic tetrahedral
and triangular finite element meshes. PhD thesis, TU Chemnitz, 1998
[KUN 00] KuNERT G., VERFURTH R., «Edge residuals dominate a posteriori
error estimates for linear finite element methods on anisotropic triangular
and tetrahedral meshes», Numer. Math., 2000, (to appear)
[LON 98] LONSING M., Zwei a posteriori Fehlerschatzer fur die Warmeleitungsgleichung.
Diplomarbeit, Ruhr-Universitat Bochum, 1998
[SIE 96] SIEBERT K. G., «An a posteriori error estimator for anisotropic refinement
», Numer. Math., vol. 73, 1996, p. 373 -398.
[VER 96] VERFURTH R., A Review of A Posteriori Error Estimation and Adaptive
Mesh - Refinement Techniques, Wiley-Teubner, Stuttgart, 1996.
[VER 98a] VERFURTH R., «Robust a posteriori error estimators for a singularly
perturbed reaction-diffusion equation», Numer. Math., vol. 78, 1998, 479 -
[VER 98b] VERFURTH R., «A posteriori error estimators for convection-diffusion
equations», Numer. Math., vol. 80, 1998, p. 641 - 663.
[VER 98c] VERFURTH R., «A posteriori error estimates for nonlinear problems.
Lr(o, T; W1·P(O))-error estimates for finite element discretizations of
parabolic equations», Numer. Meth. for PDE, vol. 14, 1998, p. 487- 518.
[VER 98d] VERFURTH R., «A posteriori error estimates for nonlinear problems.
Lr (0, T; LP(O) )-error estimates for finite element discretizations of parabolic
equations», Math. Comput., vol. 67, 1998, p. 1335- 1360.
[VER 99] VERFURTH R., «Error estimates for some quasi-interpolation operators
», Modelisation Mathematique et Analyse Numerique, vol. 33, 1999, p. 695
-713.
[VER 00] VERFURTH R., «On the constants in some inverse inequalities for
finite element functions», Modelisation Mathematique et Analyse Numerique
(submitted)