A Nitsche-eXtended Finite Element Method for Distributed Optimal Control Problems of Elliptic Interface Equations

References

[1] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. 10.1137/0719052Search in Google Scholar

[2] I. Babuška, The finite element method for elliptic equations with discontinuous coefficients, Computing 5 (1970), no. 3, 207–213. 10.1007/BF02248021Search in Google Scholar

[3] I. Babuška and U. Banerjee, Stable generalized finite element method (SGFEM), Comput. Methods Appl. Mech. Engrg. 201/204 (2012), no. 1, 91–111. 10.1016/j.cma.2011.09.012Search in Google Scholar

[4] I. Babuška, G. Caloz and J. E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31 (1994), no. 4, 945–981. 10.1137/0731051Search in Google Scholar

[5] N. Barrau, R. Becker, E. Dubach and R. Luce, A robust variant of NXFEM for the interface problem, C. R. Math. Acad. Sci. Paris 350 (2012), no. 15–16, 789–792. 10.1016/j.crma.2012.09.018Search in Google Scholar

[6] J. W. Barrett and C. M. Elliott, Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA J. Numer. Anal. 7 (1987), no. 3, 283–300. 10.1093/imanum/7.3.283Search in Google Scholar

[7] R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 41–44, 3352–3360. 10.1016/j.cma.2009.06.017Search in Google Scholar

[8] R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim. 39 (2000), no. 1, 113–132. 10.1137/S0363012999351097Search in Google Scholar

[9] J. H. Bramble and J. T. King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math. 6 (1996), no. 1, 109–138. 10.1007/BF02127700Search in Google Scholar

[10] E. Burman, S. Claus, P. Hansbo, M. G. Larson and A. Massing, CutFEM: Discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg. 104 (2015), no. 7, 472–501. 10.1002/nme.4823Search in Google Scholar

[11] Z. Cai, C. He and S. Zhang, Discontinuous finite element methods for interface problems: Robust a priori and a posteriori error estimates, SIAM J. Numer. Anal. 55 (2017), no. 1, 400–418. 10.1137/16M1056171Search in Google Scholar

[12] B. Camp, T. Lin, Y. Lin and W. Sun, Quadratic immersed finite element spaces and their approximation capabilities, Adv. Comput. Math. 24 (2006), no. 1–4, 81–112. 10.1007/s10444-004-4139-8Search in Google Scholar

[13] Y. Chen, Y. Huang, W. Liu and N. Yan, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput. 42 (2010), no. 3, 382–403. 10.1007/s10915-009-9327-8Search in Google Scholar

[14] Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998), no. 2, 175–202. 10.1007/s002110050336Search in Google Scholar

[15] K. W. Cheng and T.-P. Fries, Higher-order XFEM for curved strong and weak discontinuities, Internat. J. Numer. Methods Engrg. 82 (2010), no. 5, 564–590. 10.1002/nme.2768Search in Google Scholar

[16] W. Gong and N. Yan, Adaptive finite element method for elliptic optimal control problems: Convergence and optimality, Numer. Math. 135 (2017), no. 4, 1121–1170. 10.1007/s00211-016-0827-9Search in Google Scholar

[17] Y. Gong and Z. Li, Immersed interface finite element methods for elasticity interface problems with non-homogeneous jump conditions, Numer. Math. Theory Methods Appl. 3 (2010), no. 1, 23–39. 10.4208/nmtma.2009.m9001Search in Google Scholar

[18] D. Han, P. Wang, X. He, T. Lin and J. Wang, A 3D immersed finite element method with non-homogeneous interface flux jump for applications in particle-in-cell simulations of plasma-lunar surface interactions, J. Comput. Phys. 321 (2016), 965–980. 10.1016/j.jcp.2016.05.057Search in Google Scholar

[19] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 47–48, 5537–5552. 10.1016/S0045-7825(02)00524-8Search in Google Scholar

[20] P. Hansbo, M. G. Larson and S. Zahedi, A cut finite element method for a Stokes interface problem, Appl. Numer. Math. 85 (2014), no. 11, 90–114. 10.1016/j.apnum.2014.06.009Search in Google Scholar

[21] X. He, T. Lin and Y. Lin, Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions, Int. J. Numer. Anal. Model. 8 (2011), no. 2, 284–301. Search in Google Scholar

[22] M. Hintermüller and R. H. W. Hoppe, Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations, SIAM J. Control Optim. 48 (2010), no. 8, 5468–5487. 10.1137/090761823Search in Google Scholar

[23] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl. 30 (2005), no. 1, 45–61. 10.1007/s10589-005-4559-5Search in Google Scholar

[24] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Math. Model. Theory Appl. 23, Springer Netherlands, Dordrecht, 2009. Search in Google Scholar

[25] M. Hinze and M. Vierling, The semi-smooth newton method for variationally discretized control constrained elliptic optimal control problems; implementation, convergence and globalization. , Optim. Methods Softw. 27 (2012), no. 6, 933–950. 10.1080/10556788.2012.676046Search in Google Scholar

[26] P. Huang, H. Wu and Y. Xiao, An unfitted interface penalty finite element method for elliptic interface problems, Comput. Methods Appl. Mech. Engrg. 323 (2017), 439–460. 10.1016/j.cma.2017.06.004Search in Google Scholar

[27] K. Kergrene, I. Babuška and U. Banerjee, Stable generalized finite element method and associated iterative schemes; application to interface problems, Comput. Methods Appl. Mech. Engrg. 305 (2016), 1–36. 10.1016/j.cma.2016.02.030Search in Google Scholar

[28] C. Lehrenfeld and A. Reusken, Analysis of a Nitsche XFEM-DG discretization for a class of two-phase mass transport problems, SIAM J. Numer. Anal. 51 (2013), no. 2, 958–983. 10.1137/120875260Search in Google Scholar

[29] C. Lehrenfeld and A. Reusken, Analysis of a high-order unfitted finite element method for elliptic interface problems, IMA J. Numer. Anal. 38 (2018), no. 3, 1351–1387. 10.1093/imanum/drx041Search in Google Scholar

[30] J. Li, J. M. Melenk, B. Wohlmuth and J. Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems, Appl. Numer. Math. 60 (2010), no. 1–2, 19–37. 10.1016/j.apnum.2009.08.005Search in Google Scholar

[31] R. Li, W. Liu, H. Ma and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim. 41 (2002), no. 5, 1321–1349. 10.1137/S0363012901389342Search in Google Scholar

[32] Z. Li and K. Ito, The Immersed Interface Method, Front. Appl. Math. 33, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2006. 10.1137/1.9780898717464Search in Google Scholar

[33] T. Lin, Y. Lin and W. Sun, Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems, Discrete Contin. Dyn. Syst. Ser. B 7 (2007), no. 4, 807–823. 10.3934/dcdsb.2007.7.807Search in Google Scholar

[34] T. Lin, Y. Lin and X. Zhang, Partially penalized immersed finite element methods for elliptic interface problems, SIAM J. Numer. Anal. 53 (2015), no. 2, 1121–1144. 10.1137/130912700Search in Google Scholar

[35] T. Lin, Q. Yang and X. Zhang, Partially penalized immersed finite element methods for parabolic interface problems, Numer. Methods Partial Differential Equations 31 (2015), no. 6, 1925–1947. 10.1002/num.21973Search in Google Scholar

[36] W. Liu, W. Gong and N. Yan, A new finite element approximation of a state-constrained optimal control problem, J. Comput. Math. 27 (2009), no. 1, 97–114. Search in Google Scholar

[37] N. Moës, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing, Internat. J. Numer. Methods Engrg. 46 (1999), no. 1, 131–150. 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-JSearch in Google Scholar

[38] S. Nicaise, Y. Renard and E. Chahine, Optimal convergence analysis for the extended finite element method, Internat. J. Numer. Methods Engrg. 86 (2011), no. 4–5, 528–548. 10.1002/nme.3092Search in Google Scholar

[39] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hamburg 36 (1971), 9–15. 10.1007/BF02995904Search in Google Scholar

[40] M. Plum and C. Wieners, Optimal a priori estimates for interface problems, Numer. Math. 95 (2003), no. 4, 735–759. 10.1007/s002110200395Search in Google Scholar

[41] A. Rösch, K. G. Siebert and S. Steinig, Reliable a posteriori error estimation for state-constrained optimal control, Comput. Optim. Appl. 68 (2017), no. 1, 121–162. 10.1007/s10589-017-9908-7Search in Google Scholar

[42] R. Schneider and G. Wachsmuth, A posteriori error estimation for control-constrained, linear-quadratic optimal control problems, SIAM J. Numer. Anal. 54 (2016), no. 2, 1169–1192. 10.1137/15M1020460Search in Google Scholar

[43] B. Schott, Stabilized cut finite element methods for complex interface coupled flow problems, PhD thesis, Technische Universität Munchen, 2017. Search in Google Scholar

[44] S. Soghrati, A. M. Aragón, C. A. Duarte and P. H. Geubelle, An interface-enriched generalized FEM for problems with discontinuous gradient fields, Internat. J. Numer. Methods Engrg. 89 (2012), no. 8, 991–1008. 10.1002/nme.3273Search in Google Scholar

[45] T. Strouboulis, I. Babuška and K. Copps, The design and analysis of the generalized finite element method, Comput. Methods Appl. Mech. Engrg. 181 (2000), no. 1–3, 43–69. 10.1016/S0045-7825(99)00072-9Search in Google Scholar

[46] F. Tröltzsch, Optimal control of partial differential equations: Theory, methods and applications, SIAM J. Control Optim. 112(2):399, 2010. 10.1090/gsm/112Search in Google Scholar

[47] D. Wachsmuth and J.-E. Wurst, Optimal control of interface problems with hp-finite elements, Numer. Funct. Anal. Optim. 37 (2016), no. 3, 363–390. 10.1080/01630563.2016.1149014Search in Google Scholar

[48] Z. Weng, J. Z. Yang and X. Lu, A stabilized finite element method for the convection dominated diffusion optimal control problem, Appl. Anal. 95 (2016), no. 12, 2807–2823. 10.1080/00036811.2015.1114606Search in Google Scholar

[49] M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), no. 1, 152–161. 10.1137/0715010Search in Google Scholar

[50] H. Wu and Y. Xiao, An unfitted hp-interface penalty finite element method for elliptic interface problems, J. Comput. Math. 37 (2019), no. 3, 316–339. 10.4208/jcm.1802-m2017-0219Search in Google Scholar

[51] J. Xu, Estimate of the convergence rate of the finite element solutions to elliptic equation of second order with discontinuous coefficients, Natural Sci. J. Xiangtan Univ. 1 (1982), 1–5. Search in Google Scholar

[52] F. W. Yang, C. Venkataraman, V. Styles and A. Madzvamuse, A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws, Commun. Comput. Phys. 21 (2017), no. 1, 65–92. 10.4208/cicp.240715.080716aSearch in Google Scholar

[53] Q. Zhang, K. Ito, Z. Li and Z. Zhang, Immersed finite elements for optimal control problems of elliptic PDEs with interfaces, J. Comput. Phys. 298 (2015), 305–319. 10.1016/j.jcp.2015.05.050Search in Google Scholar