Mesh modification for evolution equations

Author:
Todd Dupont

Journal:
Math. Comp. **39** (1982), 85-107

MSC:
Primary 65M60

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658215-0

MathSciNet review:
658215

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Finite element methods for which the underlying function spaces change with time are studied. The error estimates produced are all in norms that are very naturally associated with the problems. In some cases the Galerkin solution error can be seen to be quasi-optimal. K. Miller's moving finite element method is studied in one space dimension; convergence is proved for the case of smooth solutions of parabolic problems. Most, but not all, of the analysis is done on linear problems. Although second order parabolic equations are emphasized, there is also some work on first order hyperbolic and Sobolev equations.

**[1]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[2]**Roger Alexander, Paolo Manselli, and Keith Miller,*Moving finite elements for the Stefan problem in two dimensions*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**67**(1979), no. 1-2, 57–61 (1980) (English, with Italian summary). MR**617276****[3]**T. B. Benjamin, J. L. Bona, and J. J. Mahony,*Model equations for long waves in nonlinear dispersive systems*, Philos. Trans. Roy. Soc. London Ser. A**272**(1972), no. 1220, 47–78. MR**0427868**, https://doi.org/10.1098/rsta.1972.0032**[4]**R. Bonnerot & P. Jamet, "A second order finite element method for the one-dimensional Stefan problem,"*Internat. J. Numer. Methods Engrg.*, v. 8, 1974, pp. 811-820.**[5]**R. Bonnerot and P. Jamet,*Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements*, J. Computational Phys.**25**(1977), no. 2, 163–181. MR**0474875****[6]**R. Bonnerot and P. Jamet,*A conservative finite element method for one-dimensional Stefan problems with appearing and disappearing phases*, J. Comput. Phys.**41**(1981), no. 2, 357–388. MR**626616**, https://doi.org/10.1016/0021-9991(81)90101-7**[7]**P. Jamet and R. Bonnerot,*Numerical solution of the Eulerian equations of compressible flow by a finite element method which follows the free boundary and the interfaces*, J. Computational Phys.**18**(1975), 21–45. MR**0381513****[8]**Jim Douglas Jr. and Todd Dupont,*Galerkin methods for parabolic equations*, SIAM J. Numer. Anal.**7**(1970), 575–626. MR**0277126**, https://doi.org/10.1137/0707048**[9]**Todd Dupont,*Galerkin methods for first order hyperbolics: an example*, SIAM J. Numer. Anal.**10**(1973), 890–899. MR**0349046**, https://doi.org/10.1137/0710074**[10]**Richard E. Ewing,*Numerical solution of Sobolev partial differential equations*, SIAM J. Numer. Anal.**12**(1975), 345–363. MR**0395265**, https://doi.org/10.1137/0712028**[11]**R. J. Gelinas, S. K. Doss, and Keith Miller,*The moving finite element method: applications to general partial differential equations with multiple large gradients*, J. Comput. Phys.**40**(1981), no. 1, 202–249. MR**611809**, https://doi.org/10.1016/0021-9991(81)90207-2**[12]**Pierre Jamet,*Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain*, SIAM J. Numer. Anal.**15**(1978), no. 5, 912–928. MR**507554**, https://doi.org/10.1137/0715059**[13]**Pierre Jamet,*Stability and convergence of a generalized Crank-Nicolson scheme on a variable mesh for the heat equation*, SIAM J. Numer. Anal.**17**(1980), no. 4, 530–539. MR**584728**, https://doi.org/10.1137/0717045**[14]**O. K. Jensen & B. A. Finlayson, "Solution of the transport equations using a moving coordinate system,"*Adv. in Water Resources*, v. 3, 1980, pp. 9-18.**[15]**Ole Krogh Jensen and Bruce A. Finlayson,*Oscillation limits for weighted residual methods applied to convective diffusion equations*, Internat. J. Numer. Methods Engrg.**15**(1980), no. 11, 1681–1689. MR**593595**, https://doi.org/10.1002/nme.1620151109**[16]**Daniel R. Lynch and William G. Gray,*Finite element simulation of flow in deforming regions*, J. Comput. Phys.**36**(1980), no. 2, 135–153. MR**579078**, https://doi.org/10.1016/0021-9991(80)90180-1**[17]**Keith Miller and Robert N. Miller,*Moving finite elements. I*, SIAM J. Numer. Anal.**18**(1981), no. 6, 1019–1032. MR**638996**, https://doi.org/10.1137/0718070

Keith Miller,*Moving finite elements. II*, SIAM J. Numer. Anal.**18**(1981), no. 6, 1033–1057. MR**638997**, https://doi.org/10.1137/0718071**[18]**Keith Miller and Robert N. Miller,*Moving finite elements. I*, SIAM J. Numer. Anal.**18**(1981), no. 6, 1019–1032. MR**638996**, https://doi.org/10.1137/0718070

Keith Miller,*Moving finite elements. II*, SIAM J. Numer. Anal.**18**(1981), no. 6, 1033–1057. MR**638997**, https://doi.org/10.1137/0718071**[19]**K. O'Neill & D. R. Lynch, "Effective and highly accurate solution of diffusion and convection-diffusion problems using moving, deforming coordinates." (To appear.)**[20]**Harvey S. Price and Richard S. Varga,*Error bounds for semidiscrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics*, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 74–94. MR**0266452****[21]**A. H. Schatz, V. C. Thomée, and L. B. Wahlbin,*Maximum norm stability and error estimates in parabolic finite element equations*, Comm. Pure Appl. Math.**33**(1980), no. 3, 265–304. MR**562737**, https://doi.org/10.1002/cpa.3160330305**[22]**Blair Swartz and Burton Wendroff,*Generalized finite-difference schemes*, Math. Comp.**23**(1969), 37–49. MR**0239768**, https://doi.org/10.1090/S0025-5718-1969-0239768-7**[23]**B. Wendroff,*Two-Fluid Models*:*A Critical Survey*, presented to EPRI workshop on basic two-phase flow modeling in reactor safety and performance, Tampa, Fla., 2/28-3/1/79; also LA-UR-79-291.

Retrieve articles in *Mathematics of Computation*
with MSC:
65M60

Retrieve articles in all journals with MSC: 65M60

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658215-0

Article copyright:
© Copyright 1982
American Mathematical Society