TY - JOUR
T1 - Adaptive Mesh Refinement for Elliptic Interface Problems Using the Non-Conforming Immersed Finite Element Method
AU - Wu , C.-T.
AU - Li , Z.
AU - Lai , M.-C.
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 466
EP - 483
PY - 2011
DA - 2011/08
SN - 8
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/696.html
KW - interface, immersed finite element, adaptive mesh.
AB - <p style="text-align: justify;">In this paper, an adaptive mesh refinement technique is developed
and analyzed for the non-conforming immersed finite element (IFE) method
proposed in [27]. The IFE method was developed for solving the second order
elliptic boundary value problem with interfaces across which the coefficient
may be discontinuous. The IFE method was based on a triangulation that
does not need to fit the interface. One of the key ideas of IFE method is to
modify the basis functions so that the natural jump conditions are satisfied
across the interface. The IFE method has shown to be order of $O(h^2)$ and $O(h)$ in $L^2$ norm and $H^1$ norm, respectively. In order to develop the adaptive
mesh refinement technique, additional priori and posterior error estimations are
derived in this paper. Our new a-priori error estimation shows that the generic
constant is only linearly proportional to ratio of the diffusion coefficient $\beta^-$ and $\beta^+$, which improves the corresponding result in [27]. We also show that a-posteriori
error estimate similar to the one obtained by Bernardi and Verfürth
[4] holds for the IFE solutions. Numerical examples support our theoretical
results and show that the adaptive mesh refinement strategy is effective for the
IFE approximation.</p>