Analysis of finite element approximations of a phase field model for two-phase fluids

Authors:
Xiaobing Feng, Yinnian He and Chun Liu

Journal:
Math. Comp. **76** (2007), 539-571

MSC (2000):
Primary 65M60, 35K55, 76D05

DOI:
https://doi.org/10.1090/S0025-5718-06-01915-6

Published electronically:
November 20, 2006

MathSciNet review:
2291827

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies a phase field model for the mixture of two immiscible and incompressible fluids. The model is described by a nonlinear parabolic system consisting of the nonstationary Stokes equations coupled with the Allen-Cahn equation through an extra phase induced stress term in the Stokes equations and a fluid induced transport term in the Allen-Cahn equation. Both semi-discrete and fully discrete finite element methods are developed for approximating the parabolic system. It is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model. Error estimates are derived for the semi-discrete method, and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.

**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.**I. BABUSKA, J. OSBORN AND J. PITKARANTA,*Analysis of mixed methods using mesh dependent norms*, Math. Comp., 35(1980), pp. 1039-1062. MR**0583486 (81m:65166)****3.**R. BECKER, X. FENG AND A. PROHL,*Finite element methods for the Ericksen-Leslie model of flow of nematic liquid crystals*(submitted).**4.**M. BERCOVIER AND O. PIRONNEAU,*Error estimates for finite element solution of the Stokes problem in the primitive variables*, Numer. Math., 33(1979), pp. 211-224. MR**0549450 (81g:65145)****5.**J. BEAR,*Dynamics of Fluids in Porous Media*, Dover Publications, Inc., New York, 1972.**6.**James H. Bramble and Jinchao Xu,*Some estimates for a weighted 𝐿² projection*, Math. Comp.**56**(1991), no. 194, 463–476. MR**1066830**, https://doi.org/10.1090/S0025-5718-1991-1066830-3**7.**Franco Brezzi and Michel Fortin,*Mixed and hybrid finite element methods*, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR**1115205****8.**G. CAGINALP,*An analysis of a phase field model of a free boundary*, Arch. Rational Mech. Anal., 92(1986), pp. 205-245. MR**0816623 (87c:80011)****9.**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****10.**I. V. Denisova and V. A. Solonnikov,*Solvability of a linearized problem on the motion of a drop in a fluid flow*, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**171**(1989), no. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsiĭ. 20, 53–65, 184 (Russian, with English summary); English transl., J. Soviet Math.**56**(1991), no. 2, 2309–2316. MR**1031984**, https://doi.org/10.1007/BF01671933**11.**D. A. EDWARDS, H. BRENNER AND D. T. WASAN,*Interfacial Transport Process and Rheology*, Butterworths/Heinemann, London, 1991.**12.**Xiaobing Feng and Andreas Prohl,*Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows*, Numer. Math.**94**(2003), no. 1, 33–65. MR**1971212**, https://doi.org/10.1007/s00211-002-0413-1**13.**Xiaobing Feng and Andreas Prohl,*Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits*, Math. Comp.**73**(2004), no. 246, 541–567. MR**2028419**, https://doi.org/10.1090/S0025-5718-03-01588-6**14.**Xiaobing Feng and Hai-jun Wu,*A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow*, J. Sci. Comput.**24**(2005), no. 2, 121–146. MR**2221163**, https://doi.org/10.1007/s10915-004-4610-1**15.**P. FIFE,*Dynamics of Internal Layers and Diffusive Interfaces*, SIAM, Philadelphia, PA, 1988. MR**0981594 (90c:80012)****16.**G. FIX,*Phase field method for free boundary problems*, in Free Boundary Problems (A. Fasano and M. Primicerio editors), Pitman, London, 1983, pp. 580-589.**17.**D. GILBARG AND N. S. TRUDINGER,*Elliptic Partial Differential Equations of Second Order*, Second Edition, Springer, New York, 2000.**18.**V. GIRAULT AND P. A. RAVIART,*Finite Element Method for Navier-Stokes Equations: Theory and Algorithms*, Springer-Verlag, Berlin, Heidelberg, New York, 1986. MR**0851383 (88b:65129)****19.**Yinnian He and Kaitai Li,*Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations*, Numer. Math.**79**(1998), no. 1, 77–106. MR**1608417**, https://doi.org/10.1007/s002110050332**20.**J. G. HEYWOOD AND R. RANNACHER,*Finite element approximation of the nonstationary Navier-Stokes problem I: Regularity of solutions and second-order error estimates for spatial discretization*, SIAM J. Numer. Anal., 19(1982), pp. 275-311. MR**0650052 (83d:65260)****21.**J. S. LANGER,*Models of patten formation in first-order phase transitions*, in Directions in Condensed Matter Physics, World Science Publishers, 1986, pp. 164-186. MR**0873138 (88a:82023)****22.**Gary M. Lieberman,*Second order parabolic differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR**1465184****23.**Fang-Hua Lin and Chun Liu,*Nonparabolic dissipative systems modeling the flow of liquid crystals*, Comm. Pure Appl. Math.**48**(1995), no. 5, 501–537. MR**1329830**, https://doi.org/10.1002/cpa.3160480503**24.**C. LIU AND S. SHKOLLER,*Variational phase field model for the mixture of two fluids*, Preprint 2001.**25.**Chun Liu and Jie Shen,*A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method*, Phys. D**179**(2003), no. 3-4, 211–228. MR**1984386**, https://doi.org/10.1016/S0167-2789(03)00030-7**26.**G. B. McFadden,*Phase-field models of solidification*, Recent advances in numerical methods for partial differential equations and applications (Knoxville, TN, 2001) Contemp. Math., vol. 306, Amer. Math. Soc., Providence, RI, 2002, pp. 107–145. MR**1940624**, https://doi.org/10.1090/conm/306/05251**27.**R. TEMAM,*Navier-Stokes Equations*, AMS Chelsea Publishing, Providence, RI, 2001. 1846644 (2002j:76001)

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65M60,
35K55,
76D05

Retrieve articles in all journals with MSC (2000): 65M60, 35K55, 76D05

Additional Information

**Xiaobing Feng**

Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996

Email:
xfeng@math.utk.edu

**Yinnian He**

Affiliation:
Faculty of Science, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China

Email:
heyn@mail.xjtu.edu.cn

**Chun Liu**

Affiliation:
Department of Mathematics, Penn State University, University Park, Pennsylvania 16802

Email:
liu@math.psu.edu

DOI:
https://doi.org/10.1090/S0025-5718-06-01915-6

Keywords:
Two phase fluids,
phase field model,
Allen-Cahn equation,
Stokes equations,
finite element methods

Received by editor(s):
April 28, 2005

Received by editor(s) in revised form:
August 10, 2005

Published electronically:
November 20, 2006

Additional Notes:
The work of the first author was partially supported by the NSF grant DMS-0410266.

The work of the second author was partially supported by the NSF of China grant #10671154

Article copyright:
© Copyright 2006
American Mathematical Society