Interpolation error-based a posteriori error estimation for hp-refinement using first and second derivative jumps

Applied Numerical Mathematics

Volumn 48, Issue 1, 2004, Pages 63-82

  Moore, Peter K ^a  

^a SOUTHERN METHODIST UNIVERSITY   (United States)

Author keywords

A posteriori error estimation; Hp adaptive refinement

Indexed keywords

ALGORITHMS; BOUNDARY ELEMENT METHOD; BOUNDARY VALUE PROBLEMS; FINITE ELEMENT METHOD; INTERPOLATION; PARTIAL DIFFERENTIAL EQUATIONS;

ADAPTIVE REFINEMENT; POSTERIORI ERROR ESTIMATION;

ERROR ANALYSIS;

EID: 0346973360     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2003.06.001     Document Type: Article

References (34)

1. Adjerid S., Flaherty J.E., Wang Y.J. A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems. Numer. Math. 65:1993;1-21.
2. Adjerid S., Babuška I., Flaherty J.E. A posteriori error estimation for the finite element method-of-lines solution of parabolic problems. Math. Model. Meth. Appl. Sci. 9:1999;261-286.
3. Adjerid S., Belguendouz B., Flaherty J.E. A posteriori error estimation for diffusion systems. SIAM J. Sci. Comput. 21:1999;728-746.
4. Ainsworth M., Oden J.T. A Posteriori Error Estimation in Finite Element Analysis. 2000;Wiley/Interscience, New York.
5. Babuška I., Feistauer M., Šolín P. On one approach to a posteriori error estimates for evolution problems solved by the methods of lines. Numer. Math. 89:(2):2001;225-256.
6. Babuška I., Yu D. Asymptotically exact a -posteriori error estimator for biquadratic elements. Finite Elem. Anal. Des. 3:1987;341-354.
7. Babuška I., Flaherty J.E., Henshaw W.D., Hopcroft J.E., Oliger J.E., Tezduyar T. Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations. 1995;Springer, Berlin.
8. Bank R.E., Weiser A. Some a posteriori error estimates for elliptic partial differential equations. Math. Comp. 44:19985;283-301.
9. Bramble J.H., Nitsche J.A., Schatz A.H. Maximum-norm interior estimates for Ritz-Galerkin methods. Math. Comp. 29:1975;677-688.
10. Brenan K.E., Campbell S.L., Petzold L.R. Numerical Solution of Initial Value Problems in Differential-Algebraic Equations. 1989;North-Holland, Amsterdam.
11. Demkowicz L., Rachowicz W., Devloo Ph. A fully automatic hp -adaptivity. TICAM Tech. Report. 01-28:2001;1-26.
12. Eriksson K., Johnson C. Adaptive finite element methods for linear elliptic problems. Math. Comp. 50:1988;361-383.
13. Eriksson K., Johnson C. Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal. 28:1991;43-77.
14. Eriksson K., Johnson C. Adaptive finite element methods for parabolic problems IV: Nonlinear problems. SIAM J. Numer. Anal. 32:1995;1729-1749.
15. Evans L.C. Partial Differential Equations. 1998;American Mathematical Society, Providence, RI.
16. Flaherty J.E., Moore P.K. Integrated space-time adaptive hp -refinement methods for parabolic systems. Appl. Numer. Math. 16:1995;317-341.
17. Lipman L., John F., Schechter M. Partial Differential Equations. 1964;Wiley/Interscience, New York.
18. Moore P.K. A posteriori error estimation with finite element semi- and fully-discrete methods for nonlinear parabolic equations in one space dimension. SIAM J. Numer. Anal. 31:1994;149-169.
19. Moore P.K. Comparison of adaptive methods for one-dimensional parabolic systems. Appl. Numer. Math. 16:1995;471-488.
20. Moore P.K. Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension. Numer. Math. 90:2001;149-177.
21. Moore P.K. Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for hp -adaptivity and grid-to-grid interpolation. Numer. Math. 94:2003;367-401.
22. P.K. Moore, Interpolation error-based a posteriori error estimation in three dimensions: A first step, Math. Comp., submitted for publication.
23. Nitsche J.A., Schatz A.H. Interior estimates for Ritz-Galerkin methods. Math. Comp. 28:1974;937-958.
24. Oden J.T., Demkowicz L., Rachowicz W., Westermann T.A. Toward a universal h-p adaptive finite element strategy, part 2. A posteriori error estimation. Comput. Meth. Appl. Mech. Engrg. 77:1989;113-180.
25. Rainville E.D. Special Functions. 1960;Chelsea, New York.
26. Schatz A.H. Pointwise error estimates, superconvergence, and extrapolation. Křížek M., Neittaanmäki P., Stenberg R. Finite Element Methods - Superconvergence, Post-Processing, and A Posteriori Error Estimates. 1998;Marcel Dekker, New York.
27. Segeth K. A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension. Numer. Math. 83:1999;455-476.
28. Shampine L.F., Gordon M.K. Computer Solution of Ordinary Differential Equations - The Initial Value Problem. 1975;Freeman, New York.
29. Szabó B. Mesh design for the p -version of the finite element method. Comput. Methods Appl. Mech. Engrg. 55:1986;181-197.
30. Szabó B., Babuška I. Finite Element Analysis. 1991;Wiley/Interscience, New York.
31. Wahlbin L.B. Superconvergence in Galerkin Finite Element Methods. 1995;Springer, Berlin.
32. Yu D. Asymptotically exact a-posteriori error estimator for elements of bi-even degree. Math. Numer. Sinica. 19:1991;89-101.
33. Yu D. Asymptotically exact a-posteriori error estimator for elements of bi-odd degree. Math. Numer. Sinica. 19:1991;307-314.
34. Zienkiewicz O.C., Zhu J.Z., Gong N.G. Effective and practical h-p -version adaptive analysis procedures for the finite element method. Internat. J. Numer. Math. Engrg. 28:1989;879-891.