×

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems. (English) Zbl 0791.65070

Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewise \(p\)-th- degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degree \(p+1\) that vanish at element ends.
Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.
Reviewer: Slimane Adjerid

MSC:

65M20 Method of lines 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
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations

Software:

DASSL
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Abromowitz, M., Stegun, I.A. (1965.): Handbook of mathematical functions. Dover, New York
[2] Adjerid, S., Flaherty, J.E. (1986): A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations. SIAM J. Numer. Anal.23, 778-796 · Zbl 0612.65071 · doi:10.1137/0723050
[3] Adjerid, S., Flaherty, J.E. (1986): A moving-mesh finite element method with local refinement for parabolic partial differential equations. Comput. Methods Appl. Mech. Eng.55, 3-26 · Zbl 0612.65070 · doi:10.1016/0045-7825(86)90083-6
[4] Adjerid, S., Flaherty, J.E. (1988): A local refinement finite element method for two-dimensional parabolic systems. SIAM J. Sci. Stat. Comput.9, 792-810 · Zbl 0659.65105 · doi:10.1137/0909053
[5] Adjerid, S., Flaherty, J.E. (1988): Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems. Numer. Math.53, 183-198 · Zbl 0628.65104 · doi:10.1007/BF01395884
[6] Adjerid, S., Flaherty, J.E. (1992): A posteriori error estimation for two-dimensional parabolic partial differential equations. (in preparation) · Zbl 0628.65104
[7] Adjerid, S., Flaherty, J.E., Moore, P.K., Wang, Y.J. (1991): High-order adaptive methods for parabolic systems. Tech. Rep. 91-25, Department Computer Science. Rensselaer Polytechnic Institute, Troy (Also, Physica D (1992) to appear) · Zbl 0790.65088
[8] Babuska, I. (1989): Personal communication
[9] Babuska, I., Dorr, M.R. (1981): Error estimates for the combined h and p versions of the finite element method. Numer. Math.37, 257-277 · Zbl 0487.65058 · doi:10.1007/BF01398256
[10] Babuska, I., Rheinboldt, W. (1978): Error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15, 736-754 · Zbl 0398.65069 · doi:10.1137/0715049
[11] Babuska, I., Yu, D. (1987): Asymptotically exact a-posteriori error estimator for biquadratic elements. Finite Elem. Anal. Des.3, 341-354 · Zbl 0619.73080 · doi:10.1016/0168-874X(87)90015-1
[12] Bank, R.E., Weiser, A. (1985): Some a posteriori error estimators for elliptic partial differential equations. Math. Comput.44, 283-302 · Zbl 0569.65079 · doi:10.1090/S0025-5718-1985-0777265-X
[13] Berzins, M. (1988): Global error estimation in the method of lines for parabolic equations. SIAM J. Sci. Stat. Comput.9, 687-703 · Zbl 0659.65081 · doi:10.1137/0909045
[14] Bieterman, M., Babuska, I. (1982): The finite element method for parabolic equations. I. a posteriori error estimation. Numer. Math.40, 339-371 · Zbl 0534.65072 · doi:10.1007/BF01396451
[15] Bieterman, M., Babuska, I. (1982): The finite element method for parabolic equations. II. a posteriori error estimation and adaptive approach. Numer. Math.40, 373-406 · Zbl 0534.65073 · doi:10.1007/BF01396452
[16] Coyle, J.M., Flaherty, J.E., Ludwig, R. (1986): On the stability of mesh equidistribution schemes for time-dependent partial differential equations. J. Comput. Phys.62, 26-39 · Zbl 0579.65122 · doi:10.1016/0021-9991(86)90098-7
[17] Devloo, J., Oden, J.T., Pattani, P. (1988): An h-p adaptive, finite element method for the numerical simulation of compressible flow. Comput. Methods Appl. Mech. Eng.70, 203-235 · Zbl 0636.76064 · doi:10.1016/0045-7825(88)90158-2
[18] Lawson, J., Berzins, M., Dew, P.M. (1991): Balancing space and time errors in the method of lines. SIAM. J. Sci. Stat. Comput.12, 573-594 · Zbl 0725.65087 · doi:10.1137/0912031
[19] Oden, J.T., Carey, G.F. (1983): Finite elements: mathematical aspects. Vol. IV, Prentice-Hall, Englewood Cliffs · Zbl 0496.65055
[20] Petzold, L.R. (1982): A description of DASSL: a differential/algebraic system solver, Rep. Sand. 82-8637. Sandia National Laboratory Livermore
[21] Thomée, V. (1980): Negative norm estimates and superconvergence in Galerkin Methods for Parabolic Problems. Math. Comp.34, 93-113 · Zbl 0454.65077
[22] Wait, R., Mitchell, A.R. (1985): Finite element analysis and applications. Wiley, Chichester · Zbl 0577.65093
[23] Szabo, B., Babuska, I. (1991): Finite element analysis. Wiley, New York
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.