# RECENT DEVELOPMENT OF IMMERSED FEM FOR ELLIPTIC AND ELASTIC INTERFACE PROBLEMS

• Accepted : 2019.06.10
• Published : 2019.06.25

#### Abstract

We survey a recently developed immersed finite element method (IFEM) for the interface problems. The IFEM uses structured grids such as uniform grids, even if the interface is a smooth curve. Instead of fitting the curved interface, the bases are modified so that they satisfy the jump conditions along the interface. The early versions of IFEM [1, 2] were suboptimal in convergence order [3]. Later, the consistency terms were added to the bilinear forms [4, 5], thus the scheme became optimal and the error estimates were proven. For elasticity problems with interfaces, we modify the Crouzeix-Raviart based element to satisfy the traction conditions along the interface [6], but the consistency terms are not needed. To satisfy the Korn's inequality, we add the stabilizing terms to the bilinear form. The optimal error estimate was shown for a triangular grid. Lastly, we describe the multigrid algorithms for the discretized system arising from IFEM. The prolongation operators are designed so that the prolongated function satisfy the flux continuity condition along the interface. The W-cycle convergence was proved, and the number of V-cycle is independent of the mesh size.

#### Acknowledgement

Supported by : National Research Foundation of Korea

#### References

1. Z. LI, T. LIN, AND X. WU, New cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik, 96 (2003), pp. 61-98. https://doi.org/10.1007/s00211-003-0473-x
2. Z. LI, T. LIN, Y. LIN, AND R. C. ROGERS, An immersed finite element space and its approximation capability, Numerical Methods for Partial Differential Equations, 20 (2004), pp. 338-367. https://doi.org/10.1002/num.10092
3. H. JI, J. CHEN, AND Z. LI, A symmetric and consistent immersed finite element method for interface problems, Journal of Scientific Computing, 61 (2014), pp. 533-557. https://doi.org/10.1007/s10915-014-9837-x
4. T. LIN, Y. LIN, AND X. ZHANG, Partially penalized immersed finite element methods for elliptic interface problems, SIAM Journal on Numerical Analysis, 53 (2015), pp. 1121-1144. https://doi.org/10.1137/130912700
5. D. Y. KWAK AND J. LEE, A modified P1-immersed finite element method, International Journal of Pure and Applied Mathematics, 104 (2015), pp. 471-494.
6. D. Y. KWAK, S. JIN, AND D. KYEONG, A stabilized P1-nonconforming immersed finite element method for the interface elasticity problems, ESAIM: Mathematical Modelling and Numerical Analysis, 51 (2017), pp. 187-207. https://doi.org/10.1051/m2an/2016011
7. I. BABUS KA, The finite element method for elliptic equations with discontinuous coefficients, Computing, 5 (1970), pp. 207-213. https://doi.org/10.1007/BF02248021
8. T. BELYTSCHKO, C. PARIMI, N. MOES, N. SUKUMAR, AND S. USUI, Structured extended finite element methods for solids defined by implicit surfaces, International journal for Numerical Methods in Engineering, 56 (2003), pp. 609-635. https://doi.org/10.1002/nme.686
9. T. BELYTSCHKO AND T. BLACK, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering, 45 (1999), pp. 601-620. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S
10. P. KRYSL AND T. BELYTSCHKO, An efficient linear-precision partition of unity basis for unstructured meshless methods, Communications in Numerical Methods in Engineering, 16 (2000), pp. 239-255. https://doi.org/10.1002/(SICI)1099-0887(200004)16:4<239::AID-CNM322>3.0.CO;2-W
11. G. LEGRAIN, N. MOES, AND E. VERRON, Stress analysis around crack tips in finite strain problems using the extended finite element method, International Journal for Numerical Methods in Engineering, 63 (2005), pp. 290-314. https://doi.org/10.1002/nme.1291
12. N. MOE S, J. DOLBOW, AND T. BELYTSCHKO, A finite element method for crack growth without remeshing, International journal for numerical methods in engineering, 46 (1999), pp. 131-150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
13. M. CROUZEIX AND P.-A. RAVIART, Conforming and nonconforming finite element methods for solving the stationary stokes equations I, Revue Francaise D'automatique Informatique Recherche Operationnelle. Mathematique, 7 (1973), pp. 33-75.
14. D. Y. KWAK, K. T. WEE, AND K. S. CHANG, An analysis of a broken P1-nonconforming finite element method for interface problems, SIAM Journal on Numerical Analysis, 48 (2010), pp. 2117-2134. https://doi.org/10.1137/080728056
15. S. H. CHOU, D. Y. KWAK, AND K. T. WEE, Optimal convergence analysis of an immersed interface finite element method, Advances in Computational Mathematics, 33 (2010), pp. 149-168. https://doi.org/10.1007/s10444-009-9122-y
16. D. N. ARNOLD, An interior penalty finite element method with discontinuous elements, SIAM journal on Numerical Analysis, 19 (1982), pp. 742-760. https://doi.org/10.1137/0719052
17. D. N. ARNOLD, F. BREZZI, B. COCKBURN, AND D. MARINI, Discontinuous Galerkin methods for elliptic problems, in Discontinuous Galerkin Methods, Springer, 2000, pp. 89-101.
18. C. DAWSON, S. SUN, AND M. F. WHEELER, Compatible algorithms for coupled flow and transport, Computer Methods in Applied Mechanics and Engineering, 193 (2004), pp. 2565-2580. https://doi.org/10.1016/j.cma.2003.12.059
19. K. S. CHANG AND D. Y. KWAK, Discontinuous bubble scheme for elliptic problems with jumps in the solution, Computer Methods in Applied Mechanics and Engineering, 200 (2011), pp. 494-508. https://doi.org/10.1016/j.cma.2010.06.029
20. I. K. GWANGHYUN JO, DO Y. KWAK, Two discontinuous bubble schemes for elliptic interface problems with jumps, preprint.
21. T. LIN AND X. ZHANG, Linear and bilinear immersed finite elements for planar elasticity interface problems, Journal of Computational and Applied Mathematics, 236 (2012), pp. 4681-4699. https://doi.org/10.1016/j.cam.2012.03.012
22. T. LIN, D. SHEEN, AND X. ZHANG, A locking-free immersed finite element method for planar elasticity interface problems, Journal of Computational Physics, 247 (2013), pp. 228-247. https://doi.org/10.1016/j.jcp.2013.03.053
23. J. H. BRAMBLE AND J. T. KING, A finite element method for interface problems in domains with smooth boundaries and interfaces, Advances in Computational Mathematics, 6 (1996), pp. 109-138. https://doi.org/10.1007/BF02127700
24. Z. CHEN AND J. ZOU, Finite element methods and their convergence for elliptic and parabolic interface problems, Numerische Mathematik, 79 (1998), pp. 175-202. https://doi.org/10.1007/s002110050336
25. J. DOUGLAS AND T. DUPONT, Interior penalty procedures for elliptic and parabolic galerkin methods, in Computing methods in applied sciences, Springer, 1976, pp. 207-216.
26. C. E. BAUMANN AND J. T. ODEN, A discontinuous hp finite element method for convectiondiffusion problems, Computer Methods in Applied Mechanics and Engineering, 175 (1999), pp. 311-341. https://doi.org/10.1016/S0045-7825(98)00359-4
27. S. H. CHOU, D. Y. KWAK, AND K. Y. KIM, Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems, Mathematics of Computation, 72 (2003), pp. 525-539.
28. S.-H. CHOU AND S. TANG, Conservative $P_1$ conforming and nonconforming Galerkin fems: effective flux evaluation via a nonmixed method approach, SIAM Journal on Numerical Analysis, 38 (2000), pp. 660-680. https://doi.org/10.1137/S0036142999361517
29. B. COURBET AND J. CROISILLE, Finite volume box schemes on triangular meshes, ESAIM: Mathematical Modelling and Numerical Analysis, 32 (1998), pp. 631-649. https://doi.org/10.1051/m2an/1998320506311
30. F. BREZZI, J. DOUGLAS JR, AND L. D. MARINI, Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik, 47 (1985), pp. 217-235. https://doi.org/10.1007/BF01389710
31. F. BREZZI AND M. FORTIN, Mixed and hybrid finite element methods, vol. 15, Springer-Verlag, New York, 1991.
32. P. A. RAVIART AND J. M. THOMAS, A mixed finite element method for 2-nd order elliptic problems, Mathematical aspects of finite element methods, (1977), pp. 292-315.
33. D. N. ARNOLD AND F. BREZZI, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, ESAIM: Mathematical Modelling and Numerical Analysis, 19 (1985), pp. 7-32. https://doi.org/10.1051/m2an/1985190100071
34. R. S. FALK, Nonconforming finite element methods for the equations of linear elasticity, Mathematics of Computation, 57 (1991), pp. 529-550. https://doi.org/10.1090/S0025-5718-1991-1094947-6
35. R. KOUHIA AND R. STENBERG, A linear nonconforming finite element method for nearly incompressible elasticity and stokes flow, Computer Methods in Applied Mechanics and Engineering, 124 (1995), pp. 195-212. https://doi.org/10.1016/0045-7825(95)00829-P
36. P. HANSBO AND M. G. LARSON, Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), pp. 63-72. https://doi.org/10.1051/m2an:2003020
37. S. C. BRENNER, Korn's inequalities for piecewise $H_1$ vector fields, Mathematics of Computation, (2004), pp. 1067-1087.
38. R. FEDORENKO, The speed of convergence of one iterative process, USSR Computational Mathematics and Mathematical Physics, 4 (1964), pp. 559-564.
39. W. HACKBUSCH, Multi-grid methods and applications, vol. 4, Springer-Verlag, Berlin, 1985.
40. S. F. MCCORMICK, Multigrid methods, vol. 3, SIAM, 1987.
41. R. NICOLAIDES, On some theoretical and practical aspects of multigrid methods, Mathematics of Computation, 33 (1979), pp. 933-952. https://doi.org/10.1090/S0025-5718-1979-0528048-4
42. J. H. BRAMBLE AND J. E. PASCIAK, New convergence estimates for multigrid algorithms, Mathematics of Computation, 49 (1987), pp. 311-329. https://doi.org/10.1090/S0025-5718-1987-0906174-X
43. S. C. BRENNER, An optimal-order multigrid method for P1 nonconforming finite elements, mathematics of computation, 52 (1989), pp. 1-15.
44. J. H. BRAMBLE, J. E. PASCIAK, AND J. XU, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Mathematics of Computation, 56 (1991), pp. 1-34. https://doi.org/10.1090/S0025-5718-1991-1052086-4
45. D. BRAESS, W. DAHMEN, AND C. WIENERS, A multigrid algorithm for the mortar finite element method, SIAM Journal on Numerical Analysis, 37 (1999), pp. 48-69. https://doi.org/10.1137/S0036142998335431
46. G. JO AND D. Y. KWAK, An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid, Computer Methods in Applied Mechanics and Engineering, (2017).
47. D. Y. K. GWANGHYUN JO, Geometric multigrid algorithms for elliptic interface problems using structured grids, Numerical Algorithms, (2018).
48. J. H. BRAMBLE AND J. E. PASCIAK The analysis of smoothers for multigrid algorithms, Mathematics of Computation, 58 (1992), pp. 467-488. https://doi.org/10.1090/S0025-5718-1992-1122058-0