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Continuously deforming finite elements. (English) Zbl 0588.76147
(From the summary.) The authors have introduced a general class of time- dependent co-ordinate transformations in a variational formulation for evolution problems. The variational problem is posed with respect to both solution and transformation field variables. An approximate analysis using finite elements is developed from the continuous variational form. Modified forms of the variational functional are considered to ensure the deforming mesh is not too irregular. ODE system integrators are utilized to integrate the resulting semidiscrete systems. In Part I the authors consider the formulation for problems in one spatial dimension and time, including, in particular, convection-dominated flows described by convection-diffusion, Burger’s and Buckley-Leverett equations. In part II the method is extended to two dimensions. Also presented are supporting numerical experiments.
Reviewer: P.Narain

MSC:
76Rxx Diffusion and convection
35Q99 Partial differential equations of mathematical physics and other areas of application
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[1] ’An adaptive computer program for the solution of Div (p (x, y) grad u) = f(x, y, u) on a polygonal region’, in Mathematics of Finite Elements and Applications: MAFELAP, 1975, (Ed.), Academic Press, New York, 1977, pp. 543-553.
[2] Babuška, Comp. Meth. Appl. Mech. Eng. 18 pp 519– (1979)
[3] Carey, Int. j. numer. methods eng. 17 pp 1717– (1981)
[4] and , ’A refinement algorithm and dynamic data structure for finite element meshes’. Report CNA TR-166, Univ. of Texas at Austin (1980).
[5] Bieterman, Numer. Math. 40 pp 339– (1982)
[6] T. J. R. Hughes (Ed.), Finite Element Methods for Convection-Dominated Flows, ASME Monograph AMD-34, 1979.
[7] Yanenko, Comp. Meth. Appl. Mech. Eng. 17/18 pp 659– (1979)
[8] and , ’A finite element solution for porous medium freezing using Hermite basis functions and a continuously deforming coordinate system’, In Numerical Methods in Thermal Problems. ( and , Eds.) Pineridge Press, Swansea, U.K., pp. 548-559 (1979).
[9] Miller, SIAM J. Numer. Anal. 18 pp 1019– (1981)
[10] Miller, SIAM J. Numer. Anal. 18 pp 1033– (1981)
[11] Douglas, SIAM J. Numer. Anal. 19 pp 871– (1982)
[12] and , Finite Elements–An Introduction, Prentice-Hall, Englewood Cliffs, N.J, 1981.
[13] Lynch, Int. j. numer. methods eng. 17 pp 81– (1981)
[14] and , ’Grading functions and mesh redistribution’, SIAM J. Numer. Anal., in press (1985). · Zbl 0577.65076
[15] ’Coordinate system control: adaptive meshes’, in Numerical Grid Generation (Ed.), Elsevier, New York, 1982, pp. 277-294.
[16] and , Finite Elements: A Second Course, Prentice-Hall, Englewood Cliffs, N.J., 1982.
[17] ’ODE solvers for use with method of lines’, in Advances in Computer Methods for Partial Differential Equations–IV ( and , Eds.), IMACS, 1981; Finite Element Methods for Convection-Dominated Flows (, Ed.), A.S.M.E. Monograph AMD-34, 1979.
[18] ’General collocation methods for time-dependent nonlinear boundary-value problems’, Soc. Petrol. Eng. J., 345-352 (1977).
[19] Carey, Comp. Meth. Appl. Mech. Eng. 22 pp 23– (1980)
[20] and , ’An analytical solution for linear waterflood including the effects of capillary pressure’, Soc. Petrol. Eng. J. (Feb. 1983).
[21] and , ’Treatment of numerical oscillations in heat and mass transfer problems with fronts’, Nat; Symp. on Heat Transfer, Univ. of Maryland, September 1981.
[22] and , ’On finite element analysis of heat transfer with phase change’, in Finite Elements in Water Resources, Pentech Press, London, 1981.
[23] Cravahlo, Int. J. Heat Mass Transfer 24 pp 1987– (1981)
[24] Carcy, Int. j. numer. methods eng. 19 pp 341– (1983)
[25] ’Continuously deforming finite elements for transport processes’, Ph.D. dissert., Univ. of Texas at Austin (1983).
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