×

Moving finite element methods for evolutionary problems. II: Applications. (English) Zbl 0665.65072

[For part I see ibid. 79, No.2, 245-269 (1988; reviewed above).]
Authors’ summary: In this, the second of two papers on the subject, we present applications of the moving finite element method to a number of test problems. Key features are linear elements, a direct approach to parallelism and node overtaking (avoiding penalty functions), rapid inversion of the mass matrix by preconditioned conjugate gradients, and explicit Euler time stepping. The resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy. The paper includes a substantial section on changes of dependent variable and front tracking techniques for nonlinear diffusion problems. Test problems include non-linear hyperbolic conservation laws and non-linear parabolic equations in one and two dimensions.
Reviewer: R.Gorenflo

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations

Citations:

Zbl 0665.65071
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baines, M. J.; Wathen, A. J., J. Comput. Phys., 78 (1988)
[2] Miller, K., SIAM J. Plum. Anal., 19, 1033 (1981)
[3] Gelinas, R.; Doss, S. K.; Miller, K., J. Comput. Phys., 10, 202 (1981)
[4] Dukowicz, J. K., J. Comput. Phys., 56, 324 (1984)
[5] Mosher, M. C., J. Comput. Phys., 57, 157 (1984)
[6] Hrymak, A. N.; Mcrae, G. J.; Westerberg, A. W., J. Comput. Phys., 63, 168 (1986)
[7] Wathen, A. J., SIAM J. Num. Anal., 23, No.4, 797 (1986)
[8] Wathen, A. J.; Baines, M. J., IMA J. Num. Anal., 5, 161 (1985)
[9] Dupont, T., Math. Comput., 39, 85 (1982)
[10] C. P. Please, University of Reading, U.K., private communication (1986).; C. P. Please, University of Reading, U.K., private communication (1986).
[11] P. K. Sweby, University of Reading, U.K., private communication (1986).; P. K. Sweby, University of Reading, U.K., private communication (1986).
[12] Concus, P.; Proskurowski, W., J. Comput. Phys., 30, 153 (1979)
[13] Wathen, A. J., (Ph.D. thesis (1984), University of Reading: University of Reading UK), (unpublished)
[14] Baines, M. J., (Numerical Analysis Report 15/85 (1985), University, of Reading: University, of Reading UK), (unpublished)
[15] Mueller, A. C.; Carey, G. F., Int. J. Numer. Methods Eng., 21, 2099 (1985)
[16] K. W. Morton, University of Reading, U.K., private communication (1982).; K. W. Morton, University of Reading, U.K., private communication (1982).
[17] Johnson, I. W., (Ph.D. thesis (1986), University of Reading: University of Reading UK), (unpublished)
[18] Ames, W. F., Nonlinear Partial Differential Equations in Engineering (1965), Academic Press: Academic Press New York · Zbl 0255.35001
[19] Tayler, A. B.; Ockendon, J. B.; Lacey, A. A., SIAM J. Appl. Math., 42, 1252 (1982)
[20] Zel’dovich, Y. A.B.; Kompaneets, A. S., Izv. Akad. Nauk. SSSR (1950)
[21] Tomoeda, K.; Mimura, M., Hiroshima Math. J., 13, 273 (1983)
[22] Meek, P.; Norbury, J., SIAM J. Num. Anal., 21, 883 (1984)
[23] Please, C. P.; Sweby, P. K., Numerical Analysis Report 5/86, ((1986), University of Reading: University of Reading UK), (unpublished)
[24] Johnson, I. W., (Numerical Analysis Report 12/85 (1985), University of Reading: University of Reading UK), (unpublished)
[25] Marshak, R., Phys. Fluids, 1, 1 (1985)
[26] Sod, G., J. Comput. Phys., 27, 1 (1978)
[27] Baines, M. J.; Wathen, A. J., Appl. Num. Math., 2, 495 (1986)
[28] P. K. Sweby, University of Reading, U.K., private communication (1986).; P. K. Sweby, University of Reading, U.K., private communication (1986).
[29] Edwards, M. G., (Numerical Analysis Report 20/85 (1985), University of Reading: University of Reading UK), (unpublished)
[30] Miller, K., (Babuška, I.; Zienkiewicz, O. C.; Gago, J.; de A. Oliveira, E. R., Accuracy Estimates and Adaptive Refinements in Finite Element Computations (1986), Wiley: Wiley New York), 325
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.