Mesh modification for evolution equations

Dupont, Todd

doi:10.1090/S0025-5718-1982-0658215-0

Mesh modification for evolution equations
HTML articles powered by AMS MathViewer

by Todd Dupont PDF

Math. Comp. 39 (1982), 85-107 Request permission

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.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
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. Nat. (8) 67 (1979), no. 1-2, 57–61 (1980) (English, with Italian summary). MR 617276
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 427868, DOI 10.1098/rsta.1972.0032

Internat. J. Numer. Methods Engrg.

R. Bonnerot and P. Jamet, Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements, J. Comput. Phys. 25 (1977), no. 2, 163–181. MR 474875, DOI 10.1016/0021-9991(77)90019-5
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, DOI 10.1016/0021-9991(81)90101-7
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. Comput. Phys. 18 (1975), 21–45. MR 381513, DOI 10.1016/0021-9991(75)90100-x
Jim Douglas Jr. and Todd Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575–626. MR 277126, DOI 10.1137/0707048
Todd Dupont, Galerkin methods for first order hyperbolics: an example, SIAM J. Numer. Anal. 10 (1973), 890–899. MR 349046, DOI 10.1137/0710074
Richard E. Ewing, Numerical solution of Sobolev partial differential equations, SIAM J. Numer. Anal. 12 (1975), 345–363. MR 395265, DOI 10.1137/0712028
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, DOI 10.1016/0021-9991(81)90207-2
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, DOI 10.1137/0715059
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, DOI 10.1137/0717045

Adv. in Water Resources

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, DOI 10.1002/nme.1620151109
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, DOI 10.1016/0021-9991(80)90180-1
Keith Miller and Robert N. Miller, Moving finite elements. I, SIAM J. Numer. Anal. 18 (1981), no. 6, 1019–1032. MR 638996, DOI 10.1137/0718070
Keith Miller and Robert N. Miller, Moving finite elements. I, SIAM J. Numer. Anal. 18 (1981), no. 6, 1019–1032. MR 638996, DOI 10.1137/0718070
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
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, DOI 10.1002/cpa.3160330305
Blair Swartz and Burton Wendroff, Generalized finite-difference schemes, Math. Comp. 23 (1969), 37–49. MR 239768, DOI 10.1090/S0025-5718-1969-0239768-7

Two-Fluid Models

A Critical Survey