×

zbMATH — the first resource for mathematics

A posteriori error estimation and adaptivity in the method of lines with mixed finite elements. (English) Zbl 1060.65642
The method of lines for a linear parabolic initial-boundary value problem is examined. A semi-discretization in space is done by a mixed finite element method. Then the author considers a fully adaptive time-stepping based on the Euler backward method. Some superconvergence results known for elliptic problems are applied to the semi-discrete scheme to obtain an a posteriori error estimate. Numerical experiments are included.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[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. · Zbl 0791.65070 · doi:10.1007/BF01385737 · eudml:133720
[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 · doi:10.1137/0909045
[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 · doi:10.1007/s002110050064
[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 · Zbl 0948.65120 · doi:10.1016/S0168-9274(99)00034-3
[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 · doi:10.1023/A:1022220219953 · eudml:33026
[6] J. Douglas, J.E. Roberts: Global estimates for mixed methods for second order elliptic problems. Math. Comp. 44(169) (1985), 39-52. · Zbl 0624.65109 · doi:10.2307/2007791
[7] R. Durán: Superconvergence for rectangular mixed finite elements. Numer. Math. 58 (1990), 2-15. · Zbl 0691.65076 · doi:10.1007/BF01385626
[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. · Zbl 0732.65093 · doi:10.1137/0728003
[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. · Zbl 0830.65094 · doi:10.1137/0732033
[10] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems III: Time steps variable in space. Manuscript.
[11] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems IV: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995), 1729-1749. · Zbl 0835.65116 · doi:10.1137/0732078
[12] K. Eriksson, C. Johnson: Adaptive finite element methods for parabolic problems V: Long-time integration. SIAM J. Numer. Anal. 32 (1995), 1750-1763. · Zbl 0835.65117 · doi:10.1137/0732079
[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. · Zbl 0909.65063 · doi:10.1137/S0036142996310216
[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 · doi:10.1137/0732001
[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. · Zbl 0701.65063 · doi:10.1137/0727019
[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. · Zbl 0884.00048
[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. · Zbl 0725.65087 · doi:10.1137/0912031
[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 · doi:10.1137/0913064
[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 · doi:10.1016/0377-0427(93)90093-Q
[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 · doi:10.1137/S0036142995280808
[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.
[22] A.H. Schatz, V. Thomeé, W.L. Wendland (eds): Mathematical Theory of Finite and Boundary Element Methods. Birkhäuser Verlag, Basel, 1990.
[23] V. Thomée: Galerkin Finite Element Methods for Parabolic Problems. Lecture Notes in Mathematics 1054, Springer Verlag, New York, 1998.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.