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Volume 12, Issue 3
An Interface-Unfitted Finite Element Method for Elliptic Interface Optimal Control Problems

Chaochao Yang, Tao Wang & Xiaoping Xie

DOI: 10.4208/nmtma.OA-2018-0031

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 727-749.

Published online: 2019-04

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  • Abstract

This paper develops and analyses numerical approximation for linear-quadratic optimal control problems governed by elliptic interface equations. We adopt variational discretization concept to discretize optimal control problems, and apply an interface-unfitted finite element method due to [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(47-48): 5537-5552, 2002] to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around interface are enriched into standard conforming finite element space. Optimal error estimates in both $L$2 norm and a mesh-dependent norm are derived for the optimal state, co-state and control under different regularity assumptions. Numerical results verify the theoretical results.

  • Keywords

Interface equations, interface control, variational discretization concept, cut finite element method.

  • AMS Subject Headings

49J20, 49M25, 65N12, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-12-727, author = {Chaochao Yang, Tao Wang and Xiaoping Xie}, title = {An Interface-Unfitted Finite Element Method for Elliptic Interface Optimal Control Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {3}, pages = {727--749}, abstract = {

This paper develops and analyses numerical approximation for linear-quadratic optimal control problems governed by elliptic interface equations. We adopt variational discretization concept to discretize optimal control problems, and apply an interface-unfitted finite element method due to [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(47-48): 5537-5552, 2002] to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around interface are enriched into standard conforming finite element space. Optimal error estimates in both $L$2 norm and a mesh-dependent norm are derived for the optimal state, co-state and control under different regularity assumptions. Numerical results verify the theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0031}, url = {http://global-sci.org/intro/article_detail/nmtma/13128.html} }
TY - JOUR T1 - An Interface-Unfitted Finite Element Method for Elliptic Interface Optimal Control Problems AU - Chaochao Yang, Tao Wang & Xiaoping Xie JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 727 EP - 749 PY - 2019 DA - 2019/04 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2018-0031 UR - https://global-sci.org/intro/article_detail/nmtma/13128.html KW - Interface equations, interface control, variational discretization concept, cut finite element method. AB -

This paper develops and analyses numerical approximation for linear-quadratic optimal control problems governed by elliptic interface equations. We adopt variational discretization concept to discretize optimal control problems, and apply an interface-unfitted finite element method due to [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(47-48): 5537-5552, 2002] to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around interface are enriched into standard conforming finite element space. Optimal error estimates in both $L$2 norm and a mesh-dependent norm are derived for the optimal state, co-state and control under different regularity assumptions. Numerical results verify the theoretical results.

Chaochao Yang, Tao Wang and Xiaoping Xie. (2019). An Interface-Unfitted Finite Element Method for Elliptic Interface Optimal Control Problems. Numerical Mathematics: Theory, Methods and Applications. 12 (3). 727-749. doi:10.4208/nmtma.OA-2018-0031
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