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.
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
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.
[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. |