A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

Derrick Jones; Xu Zhang; Derrick Jones; Xu Zhang

doi:10.3934/era.2021032

Electronic Research Archive

2021, Volume 29, Issue 5: 3171-3191. doi: 10.3934/era.2021032

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A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

Derrick Jones ^{1
,},
Xu Zhang ^{2
,
,}

1.
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA
2.
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA

Received: 01 November 2020 Revised: 01 February 2021 Published: 15 April 2021
Primary: 65N15, 65N30; Secondary: 35R05

In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise constant. Unisolvency, among other fundamental properties of the new vector-valued IFE functions, is analyzed. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. Re-meshing is not required in our numerical scheme for solving the moving-interface problem. Numerical experiments are carried out to demonstrate the performance of this new IFE method.
- Conforming-nonconforming immersed finite element,
- Stokes interface problem,
- moving interface
Citation: Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces[J]. Electronic Research Archive, 2021, 29(5): 3171-3191. doi: 10.3934/era.2021032

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Abstract

In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise constant. Unisolvency, among other fundamental properties of the new vector-valued IFE functions, is analyzed. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. Re-meshing is not required in our numerical scheme for solving the moving-interface problem. Numerical experiments are carried out to demonstrate the performance of this new IFE method.

References

[1]	An immersed discontinuous finite element method for Stokes interface problems. Comput. Methods Appl. Mech. Engrg. (2015) 293: 170-190.
[2]	On nonconforming linear-constant elements for some variants of the Stokes equations. Istit. Lombardo Accad. Sci. Lett. Rend. A (1993) 127: 83-93.
[3]	N. Chaabane, Immersed and Discontinuous Finite Element Methods, Thesis (Ph.D.)-Virginia Polytechnic Institute and State University. 2015.
[4]	Z. Chen, Finite Element Methods and their Applications, Scientific Computation. Springer-Verlag, Berlin, 2005.
[5]	A $P_2$-$P_1$ partially penalized immersed finite element method for Stokes interface problems. Int. J. Numer. Anal. Model. (2021) 18: 120-141.
[6]	Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Franç caise Automat. Informat. Recherche Opérationnelle Sér. Rouge (1973) 7: 33-75.
[7]	Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries. Comput. Methods Appl. Mech. Engrg. (2004) 193: 4819-4836.
[8]	V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, volume 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. Theory and algorithms. doi: 10.1007/978-3-642-61623-5
[9]	An extended pressure finite element space for two-phase incompressible flows with surface tension. J. Comput. Phys. (2007) 224: 40-58.
[10]	Solving parabolic moving interface problems with dynamical immersed spaces on unfitted meshes: Fully discrete analysis. SIAM J. Numer. Anal. (2021) 59: 797-828.
[11]	A group of immersed finite element spaces for elliptic interface problems. IMA J. Numer. Anal. (2019) 39: 482-511.
[12]	A fixed mesh method with immersed finite elements for solving interface inverse problems. J. Sci. Comput. (2019) 79: 148-175.
[13]	R. Guo, T. Lin and Y. Lin, Recovering elastic inclusions by shape optimization methods with immersed finite elements, J. Comput. Phys., 404 (2020), 109123, 24 pp. doi: 10.1016/j.jcp.2019.109123
[14]	Improved error estimation for the partially penalized immersed finite element methods for elliptic interface problems. Int. J. Numer. Anal. Model. (2019) 16: 575-589.
[15]	A cut finite element method for a Stokes interface problem. Appl. Numer. Math. (2014) 85: 90-114.
[16]	Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions. Int. J. Numer. Anal. Model. (2011) 8: 284-301.
[17]	Immersed finite element methods for parabolic equations with moving interface. Numer. Methods Partial Differential Equations (2013) 29: 619-646.
[18]	Residual-based a posteriori error estimation for immersed finite element methods. J. Sci. Comput. (2019) 81: 2051-2079.
[19]	V. John, Finite Element Methods for Incompressible Flow Problems, volume 51 of Springer Series in Computational Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-45750-5
[20]	D. Jones and X. Zhang, A class of nonconforming immersed finite element methods for Stokes interface problems, J. Comput. Appl. Math., 392 (2021), 113493. doi: 10.1016/j.cam.2021.113493
[21]	A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. (1995) 124: 195-212.
[22]	R. Lan and P. Sun, A monolithic arbitrary Lagrangian-Eulerian finite element analysis for a Stokes/parabolic moving interface problem, J. Sci. Comput., 82 (2020), Paper No. 59, 36 pp. doi: 10.1007/s10915-020-01161-9
[23]	A new local stabilized nonconforming finite element method for the Stokes equations. Computing (2008) 82: 157-170.
[24]	New Cartesian grid methods for interface problems using the finite element formulation. Numer. Math. (2003) 96: 61-98.
[25]	A method of lines based on immersed finite elements for parabolic moving interface problems. Adv. Appl. Math. Mech. (2013) 5: 548-568.
[26]	Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal. (2015) 53: 1121-1144.
[27]	A locking-free immersed finite element method for planar elasticity interface problems. J. Comput. Phys. (2013) 247: 228-247.
[28]	A nonconforming immersed finite element method for elliptic interface problems. J. Sci. Comput. (2019) 79: 442-463.
[29]	Linear and bilinear immersed finite elements for planar elasticity interface problems. J. Comput. Appl. Math. (2012) 236: 4681-4699.
[30]	T. Lin and Q. Zhuang, Optimal error bounds for partially penalized immersed finite element methods for parabolic interface problems, J. Comput. Appl. Math., 366 (2020), 112401, 11 pp. doi: 10.1016/j.cam.2019.112401
[31]	Distributed Lagrange multiplier-fictitious domain finite element method for Stokes interface problems. Int. J. Numer. Anal. Model. (2019) 16: 939-963.
[32]	Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equations (1992) 8: 97-111.
[33]	B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, volume 35 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. Theory and implementation. doi: 10.1137/1.9780898717440
[34]	Fictitious domain finite element method for Stokes/elliptic interface problems with jump coefficients. J. Comput. Appl. Math. (2019) 356: 81-97.
[35]	A numerical solution of the Navier-Stokes equations using the finite element technique. Internat. J. Comput. & Fluids (1973) 1: 73-100.
[36]	A nonconforming Nitsche's extended finite element method for Stokes interface problems. J. Sci. Comput. (2019) 81: 342-374.
[37]	N. K. Yamaleev, D. C. Del Rey Fernández, J. Lou and M. H. Carpenter, Entropy stable spectral collocation schemes for the 3-D Navier-Stokes equations on dynamic unstructured grids, J. Comput. Phys., 399 (2019), 108897, 27 pp. doi: 10.1016/j.jcp.2019.108897
[38]	A 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations. Int. J. Numer. Anal. Model. (2017) 14: 730-743.

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