×

A novel arbitrary Lagrangian-Eulerian finite element method for a parabolic/mixed parabolic moving interface problem. (English) Zbl 1479.65007

This article discusses a monolithic arbitrary Lagrangian-Eulerian (ALE) finite element method. A stable Stokes-pair mixed finite element within a specific stabilization technique and a novel ALE time-difference scheme are developed to discretize this interface problem in both semi-discrete and fully discrete fashion, for which the stability and error estimate analyses are conducted. Numerical experiments are carried out to validate all theoretical results in different cases.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
74S05 Finite element methods applied to problems in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Biot, M. A., Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., 25, 182-185 (1955) · Zbl 0067.23603
[2] Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164 (1941) · JFM 67.0837.01
[3] Atalla, N.; Panneton, R.; Debergue, P., A mixed displacement-pressure formulation for poroelastic materials, J. Acoust. Soc. Am., 104, 1444-1452 (1998)
[4] Chen, J., Time domain fundamental solution to Biot’s complete equations of dynamic poroelasticity. Part II: Three-dimensional solution, Int. J. Solids Struct., 31, 169-202 (1994) · Zbl 0816.73001
[5] Hirth, C.; Amsden, A. A.; Cook, J., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 14, 3, 227-253 (1974) · Zbl 0292.76018
[6] Hughes, T. J.; Liu, W. K.; Zimmermann, T., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 29, 3, 329-349 (1981) · Zbl 0482.76039
[7] Huerta, A.; Liu, W. K., Viscous flow structure interaction, J. Press. Vessel Technol., 110, 1, 15-21 (1988)
[8] Nitikitpaiboon, C.; Bathe, K. J., An arbitrary Lagrangian-Eulerian velocity potential formulation for fluid-structure interaction, Comput. Struct., 47, 4, 871-891 (1993) · Zbl 0800.73296
[9] (Souli, M.; Benson, D. J., Arbitrary Lagrangian Eulerian and Fluid-Structure Interaction: Numerical Simulation (2010), John Wiley & Sons)
[10] Gastaldi, L., A priori error estimates for the arbitrary Lagrangian Eulerian formulation with finite elements, East-West J. Numer. Math., 9, 123-156 (2001) · Zbl 0988.65082
[11] Formaggia, L.; Nobile, F., A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements, East-West J. Numer. Math., 7, 105-132 (1999) · Zbl 0942.65113
[12] Martín, J. S.; Smaranda, L.; Takahashi, T., Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time, J. Comput. Appl. Math., 230, 521-545 (2009) · Zbl 1166.76029
[13] Lan, R.; Ramirez, M.; Sun, P., Finite element analysis of an arbitrary Lagrangian-Eulerian method for Stokes/parabolic moving interface problem with jump coefficients, Results Appl. Math., 8, 100091 (2020) · Zbl 1443.74261
[14] Brezzi, F.; Fortin, M.; Marini, L. D., Mixed finite element methods with continuous stresses, Math. Models Methods Appl. Sci., 3, 02, 275-287 (1993) · Zbl 0774.73066
[15] Yang, D., A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media, Numer. Methods Partial Differential Equations, 17, 229-249 (2001) · Zbl 1008.76044
[16] Boffi, D.; Gastaldi, L., Stability and geometric conservation laws for ALE formulations, Comput. Methods Appl. Mech. Engrg., 193, 42-44, 4717-4739 (2004) · Zbl 1112.76382
[17] Rognes, M. E.; Kirby, R. C.; Logg, A., Efficient assembly of H(div) and H(curl) conforming finite elements, SIAM J. Sci. Comput., 31, 4130-4151 (2010) · Zbl 1206.65248
[18] Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications, Vol. 44 (2013), Springer · Zbl 1277.65092
[19] Reynolds, O., Papers on Mechanical and Physical Subjects: The Sub-Mechanics of the Universe, Vol. 3 (1903), Cambridge University Press: Cambridge University Press Cambridge
[20] Leal, L. G., Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes (2007), Cambridge University Press · Zbl 1133.76001
[21] Ciarlet, P. G., Finite Element Method for Elliptic Problems (2002), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA
[22] Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 337-344 (1984) · Zbl 0593.76039
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.