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A moving finite element method for the solution of two-dimensional time-dependent models. (English) Zbl 1013.65107
Summary: A moving finite element method is developed to solve time dependent problems in two space dimensions. In this formulation the solution is approximated by a piecewise polynomial of high degree on a hexagonally connected triangular mesh. Special treatment, such as the way of calculating the integrals involving second spatial derivatives and the way of preventing singularities are introduced and discussed. Numerical experiments are employed to illustrate the relationship between the nodes movements and the choice of penalty constants as well as to test the accuracy and efficiency of the proposed method.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Software:
LSODE; NKA
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References:
[1] Alexander, R.; Manselli, P.; Miller, K., Moving finite elements for the Stefan problem in two dimension, Rend. accad. naz. lincei (Rome), 57-61, (1979) · Zbl 0493.65068
[2] Baines, M.J., Moving finite elements, (1994), Oxford University Press Oxford · Zbl 0817.65082
[3] Baines, M.J., Grid adaptation via node movement, Appl. numer. math., 1-2, 77-96, (1998) · Zbl 0889.65122
[4] Carlson, N.; Miller, K., The gradient-weighted moving finite element method in two dimension, (), 152-164
[5] Carlson, N.; Miller, K., Design and application of a gradient-weighted moving finite element code II: in two dimension, SIAM J. sci. comput., 19, 3, 766-798, (1998) · Zbl 0911.65088
[6] Coimbra, M.C.; Sereno, C.; Rodrigues, A., Moving finite elements: simulation of a pressurization of adsorption beds with a mixture of helium-methane, (), 9-17 · Zbl 0900.76282
[7] Coimbra, M.C.; Sereno, C.; Rodrigues, A., Modelling multicomponent adsorption process by a moving finite elements method, J. comput. appl. math., 115, 169-179, (2000) · Zbl 0947.65110
[8] Coimbra, M.C.; Sereno, C.; Rodrigues, A., Applications of a moving finite element method, Chemical engrg. J., 84, 23-29, (2001)
[9] Gelinas, R.J.; Doss, S.K.; Vajk, J.P.; Djomehri, J.; Miller, K., Moving finite elements in 2D: fluid dynamics examples, (), 29-36
[10] Hansen, J.A.; Hassager, O., A new moving finite element method based on quadratic approximation functions, Internat. J. numer. meth. engrg., 28, 415-430, (1989) · Zbl 0672.73060
[11] Hindmarsh, A.C., LSODE and LSODI, two new initial value ordinary differential equation solvers, ACM-SIGNUM newslett, 15, 10-11, (1980)
[12] Jimack, P.K.; Wathen, A.J., Temporal derivatives in finite-element method on continuously deforming grids, SIAM J. numer. anal., 28, 4, 990-1003, (1991) · Zbl 0747.65083
[13] Jimack, P.K., A best approximation propriety of the moving finite element method, SIAM J. numer. anal., 33, 6, 2286-2302, (1996) · Zbl 0863.65060
[14] Johnson, I.W.; Wathen, A.; Baines, M.J., Moving finite elements for evolutionary problems (II) applications, J. comput. phys., 79, 270-297, (1988) · Zbl 0665.65072
[15] Miller, K.; Miller, R.N., Moving finite elements, part II, SIAM J. numer. anal., 18, 1033-1057, (1981) · Zbl 0518.65083
[16] Miller, K., Moving finite elements, part I, SIAM J. numer. anal., 18, 1019-1032, (1981) · Zbl 0518.65082
[17] Mueller, A.C.; Carey, G.F., Continously deforming finite elements, Internat. J. numer. meth. engrg., 19, 2099-2126, (1985) · Zbl 0588.76147
[18] C. Sereno, Método dos elementos finitos móveis aplicações em engenharia química., Ph.D. Thesis, University of Porto, Porto, 1989
[19] Sereno, C.; Rodrigues, A.; Villadsen, J., The moving finite element method with polynomial approximation of any degree, Comput. chem. engrg., 15, 25-33, (1991)
[20] Sereno, C.; Rodrigues, A.; Villadsen, J., Solution of partial differential equations systems by the moving finite element method, Comput. chem. engng., 16, 583-592, (1992)
[21] Zegeling, P., Moving finite element solution of time-dependent partial differential equation in two space dimensions, Comput. fuid dyn., 1, 135-159, (1993)
[22] Zegeling, P., r-refinement for evolutionary PDEs with finite elements or finite differences, Appl. numer. math., 1-2, 97-104, (1998) · Zbl 0891.65101
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