Previous |  Up |  Next


grid adjustment; principle of equidistribution of monitor; a posteriori error estimate; parabolic equation; finite element method; method of lines
The adjustment of one-dimensional space grid for a parabolic partial differential equation solved by the finite element method of lines is considered in the paper. In particular, the approach based on a posteriori error indicators and error estimators is studied. A statement on the rate of convergence of the approximation of error by estimator to the error in the case of a system of parabolic equations is presented.
[1] S. Adjerid J. E. Flaherty: A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations. SIAM J. Numer. Anal. 23 (1986), 778-796. DOI 10.1137/0723050 | MR 0849282
[2] S. Adjerid J.E. Flaherty Y.J. Wang: A Posteriori Error Estimation with Finite Element Methods of Lines for One-Dimensional Parabolic Systems. Tech. Report 91-1, Troy, NY, Dept. of Computer Science, Rensselaer Polytechnic Institute, 1991. MR 1217436
[3] I. Babuška W. Gui: Basic principles of feedback and adaptive approaches in the finite element method. Comput. Methods Appl. Mech. Engrg. 55 (1986), 27-42. DOI 10.1016/0045-7825(86)90084-8 | MR 0845412
[4] I. Babuška W.C. Rheinboldt: A-posteriori error estimates for the finite element method. Internat. J. Numer. Methods Engrg. 12 (1978), 1597-1615. DOI 10.1002/nme.1620121010
[5] M. Bieterman I. Babuška: An adaptive method of lines with error control for parabolic equations of the reaction-diffusion type. J. Comput. Phys. 63 (1986), 33-66. DOI 10.1016/0021-9991(86)90083-5 | MR 0832563
[6] R. M. Furzeland J. G. Verwer P. A. Zegeling: A numerical study of three moving grid methods for one-dimensional partial differential equations which are based on the method of lines. J. Comput. Phys. 89 (1990), 349-388. DOI 10.1016/0021-9991(90)90148-T | MR 1067050
[7] B. M. Herbst S. W. Schoombie A. R. Mitchell: Equidistributing principles in moving finite element methods. J. Comput. Appl. Math. 9 (1983), 377-389. DOI 10.1016/0377-0427(83)90009-2 | MR 0729241
[8] A. C. Hindmarsh: LSODE and LSODI, two new initial value ordinary differential equation solvers. ACM SIGNUM Newsletter 15 (1980), 10-11. DOI 10.1145/1218052.1218054
[9] J. Hugger: Density Representation of Finite Element Meshes for One and Two Dimensional Problems, Non-Singular or with Point Singularities. Part 1 and 2, Preprint, College Park, MD, IPST, University of Maryland, 1992. MR 1195582
[10] K. Miller: Moving finite elements II. SIAM J. Numer. Anal. 18 (1981), 1033-1057. DOI 10.1137/0718071 | MR 0638997 | Zbl 0518.65083
[11] K. Miller R. N. Miller: Moving finite elements I. SIAM J. Numer. Anal. 18 (1981), 1019-1032. DOI 10.1137/0718070 | MR 0638996
[12] J. T. Oden G. F. Carey: Finite Elements: Mathematical Aspects, Vol. IV. Englewood Cliffs, NJ, Prentice-Hall, 1983. MR 0767806
[13] L. R. Petzold: A Description of DDASSL: A Differential/Algebraic System Solver. Sandia Report No. Sand 82-8637, Livermore, CA, Sandia National Laboratory, 1982. MR 0751605
[14] Y. Ren R. D. Russell: Moving Mesh Techniques Based upon Equidistribution, and Their Stability. Preprint, Burnaby, B.C., Dept. of Mathematics and Statistics, Simon Fraser University, 1989. MR 1185646
[15] V. Thomée: Negative norm estimates and superconvergence in Galerkin methods for parabolic problems. Math. Соmр. 34 (1980), 93-113. MR 0551292
[16] R. Wait A. R. Mitchell: Finite Element Analysis and Applications. Chichester, J. Wiley and Sons, 1985. MR 0817440
Partner of
EuDML logo