| Title:
|
A posteriori error estimation and adaptivity in the method of lines with mixed finite elements (English) |
| Author:
|
Brandts, Jan H. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
44 |
| Issue:
|
6 |
| Year:
|
1999 |
| Pages:
|
407-419 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We will investigate the possibility to use superconvergence results for the mixed finite element discretizations of some time-dependent partial differential equations in the construction of a posteriori error estimators. Since essentially the same approach can be followed in two space dimensions, we will, for simplicity, consider a model problem in one space dimension. (English) |
| Keyword:
|
superconvergence |
| Keyword:
|
method of lines |
| Keyword:
|
mixed finite elements |
| Keyword:
|
a posteriori error estimation |
| Keyword:
|
adaptive time-stepping |
| Keyword:
|
adaptive refinement |
| MSC:
|
65M15 |
| MSC:
|
65M20 |
| MSC:
|
65M60 |
| idZBL:
|
Zbl 1060.65642 |
| idMR:
|
MR1727979 |
| DOI:
|
10.1023/A:1022268703907 |
| . |
| Date available:
|
2009-09-22T18:01:37Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134418 |
| . |
| Reference:
|
[1] S. Adjerid, J.E. Flaherty, Y.J. Wang: A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems.Numer. Math. 65 (1993), 1–21. MR 1217436, 10.1007/BF01385737 |
| Reference:
|
[2] M. Berzins: Global error estimation in the method of lines for parabolic equations.SIAM J. Sci. Stat. Comput. 9(4) (1988), 687–703. Zbl 0659.65081, MR 0945932, 10.1137/0909045 |
| Reference:
|
[3] J.H. Brandts: Superconvergence and a posteriori error estimation for triangular mixed finite elements.Numer. Math. 68(3) (1994), 311–324. Zbl 0823.65103, MR 1313147, 10.1007/s002110050064 |
| Reference:
|
[4] J.H. Brandts: Superconvergence for triangular order $k=1$ Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods.Appl. Numer. Math. (1996), to appear (accepted). Zbl 0948.65120, MR 1755693 |
| Reference:
|
[5] J.H. Brandts: Superconvergence of mixed finite element semi-discretizations of two time-dependent problems.Appl. Math. 44(1) (1999), 43–53. Zbl 1059.65518, MR 1666846, 10.1023/A:1022220219953 |
| Reference:
|
[6] J. Douglas, J.E. Roberts: Global estimates for mixed methods for second order elliptic problems.Math. Comp. 44(169) (1985), 39–52. MR 0771029, 10.1090/S0025-5718-1985-0771029-9 |
| Reference:
|
[7] R. Durán: Superconvergence for rectangular mixed finite elements.Numer. Math. 58 (1990), 2–15. MR 1075159, 10.1007/BF01385626 |
| Reference:
|
[8] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems I: A linear model problem.SIAM J. Numer. Anal. 28 (1991), 43–77. MR 1083324, 10.1137/0728003 |
| Reference:
|
[9] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems II: Optimal error estimates in $L_{\infty }L_2$ and $L_{\infty }L_{\infty }$.SIAM J. Numer. Anal. 32 (1995), 706–740. MR 1335652, 10.1137/0732033 |
| Reference:
|
[10] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems III: Time steps variable in space.Manuscript. |
| Reference:
|
[11] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems IV: Nonlinear problems.SIAM J. Numer. Anal. 32 (1995), 1729–1749. MR 1360457, 10.1137/0732078 |
| Reference:
|
[12] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems V: Long-time integration.SIAM J. Numer. Anal. 32 (1995), 1750–1763. MR 1360458, 10.1137/0732079 |
| Reference:
|
[13] K. Eriksson, C. Johnson, S. Larsson: Adaptive finite element methods for parabolic problems VI: Analytic semigroups.SIAM J. Numer. Anal. 35(4) (1998), 1315–1325. MR 1620144, 10.1137/S0036142996310216 |
| Reference:
|
[14] D. Estep: A posteriori error bounds and global error control for approximation of ordinary differential equations.SIAM J. Numer. Anal. 32(1) (1995), 1–48. Zbl 0820.65052, MR 1313704, 10.1137/0732001 |
| Reference:
|
[15] C. Johnson, Y. Nie, V. Thomée: An a posteriori error estimate and adaptive time step control for a backward Euler discretization of a parabolic problem.SIAM J. Numer. Anal. 27(2) (1990), 277–291. MR 1043607, 10.1137/0727019 |
| Reference:
|
[16] M. Křížek, P. Neittaanmäki, R. Stenberg (eds): Finite element methods: superconvergence, post-processing and a posteriori estimates.Proc. Conf. Univ. of Jyväskylä, 1996, Lecture Notes in Pure and Applied Mathematics volume 196, Marcel Dekker, New York, 1998. MR 1602809 |
| Reference:
|
[17] J. Lawson, M. Berzins, P.M. Dew: Balancing space and time errors in the method of lines for parabolic equations.SIAM J. Sci. Stat. Comput. 12(3) (1991), 573–594. MR 1093207, 10.1137/0912031 |
| Reference:
|
[18] P. Monk: A comparison of three mixed methods for the time-dependent Maxwell’s equations.SIAM J. Sci. Stat. Comput. 13(5) (1992), 1097–1122. Zbl 0762.65081, MR 1177800, 10.1137/0913064 |
| Reference:
|
[19] P. Monk: An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations.J. Comp. Appl. Math. 47 (1993), 101–121. Zbl 0784.65091, MR 1226366, 10.1016/0377-0427(93)90093-Q |
| Reference:
|
[20] A.K. Pani: An $H^1$-Galerkin mixed finite element method for parabolic partial differential equations.SIAM J. Numer. Anal. 35(2) (1998), 712–727. Zbl 1096.76516, MR 1618886, 10.1137/S0036142995280808 |
| Reference:
|
[21] P.A. Raviart, J.M. Thomas: A mixed finite element method for second order elliptic problems.Lecture Notes in Mathematics 606, 1977, pp. 292–315. MR 0483555 |
| Reference:
|
[22] A.H. Schatz, V. Thomeé, W.L. Wendland (eds): Mathematical Theory of Finite and Boundary Element Methods.Birkhäuser Verlag, Basel, 1990. MR 1116555 |
| Reference:
|
[23] V. Thomée: Galerkin Finite Element Methods for Parabolic Problems.Lecture Notes in Mathematics 1054, Springer Verlag, New York, 1998. MR 0744045 |
| . |
