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Volume 11, Issue 3
Higher Degree Immersed Finite Element Methods for Second-Order Elliptic Interface Problems

S. Adjerid, M. Ben-Romdhane & T. Lin

Int. J. Numer. Anal. Mod., 11 (2014), pp. 541-566.

Published online: 2014-11

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  • Abstract

We present higher degree immersed finite element (IFE) spaces that can be used to solve two dimensional second order elliptic interface problems without requiring the mesh to be aligned with the material interfaces. The interpolation errors in the proposed piecewise $p^{th}$ degree spaces yield optimal $\mathcal{O}(h^{p+1})$ and $\mathcal{O}(h^p)$ convergence rates in the $L^2$ and broken $H^1$ norms, respectively, under mesh refinement. A partially penalized method is developed which also converges optimally with the proposed higher degree IFE spaces. While this penalty is not needed when either linear or bilinear IFE space is used, a numerical example is presented to show that it is necessary when a higher degree IFE space is used.

  • Keywords

Immersed finite element, immersed interface, interface problems, Cartesian mesh method, structured mesh, higher degree finite element.

  • AMS Subject Headings

65N15, 65N30, 65N50, 35R05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-541, author = {S. Adjerid, M. Ben-Romdhane and T. Lin}, title = {Higher Degree Immersed Finite Element Methods for Second-Order Elliptic Interface Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {3}, pages = {541--566}, abstract = {

We present higher degree immersed finite element (IFE) spaces that can be used to solve two dimensional second order elliptic interface problems without requiring the mesh to be aligned with the material interfaces. The interpolation errors in the proposed piecewise $p^{th}$ degree spaces yield optimal $\mathcal{O}(h^{p+1})$ and $\mathcal{O}(h^p)$ convergence rates in the $L^2$ and broken $H^1$ norms, respectively, under mesh refinement. A partially penalized method is developed which also converges optimally with the proposed higher degree IFE spaces. While this penalty is not needed when either linear or bilinear IFE space is used, a numerical example is presented to show that it is necessary when a higher degree IFE space is used.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/541.html} }
TY - JOUR T1 - Higher Degree Immersed Finite Element Methods for Second-Order Elliptic Interface Problems AU - S. Adjerid, M. Ben-Romdhane & T. Lin JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 541 EP - 566 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/541.html KW - Immersed finite element, immersed interface, interface problems, Cartesian mesh method, structured mesh, higher degree finite element. AB -

We present higher degree immersed finite element (IFE) spaces that can be used to solve two dimensional second order elliptic interface problems without requiring the mesh to be aligned with the material interfaces. The interpolation errors in the proposed piecewise $p^{th}$ degree spaces yield optimal $\mathcal{O}(h^{p+1})$ and $\mathcal{O}(h^p)$ convergence rates in the $L^2$ and broken $H^1$ norms, respectively, under mesh refinement. A partially penalized method is developed which also converges optimally with the proposed higher degree IFE spaces. While this penalty is not needed when either linear or bilinear IFE space is used, a numerical example is presented to show that it is necessary when a higher degree IFE space is used.

S. Adjerid, M. Ben-Romdhane and T. Lin. (2014). Higher Degree Immersed Finite Element Methods for Second-Order Elliptic Interface Problems. International Journal of Numerical Analysis and Modeling. 11 (3). 541-566. doi:
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