## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - B
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- Journal of Dynamics & Games
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DCDS-B

In this article, a locally stabilized finite element formulation
of the two-dimensional Navier-Stokes problem is used.
A macroelement condition which provides the stability
of the $Q_1-P_0$ quadrilateral element and
the $P_1-P_0$ triangular element is introduced.
Moreover, the $H^1$ and $L^2$-error estimates of optimal order
for finite element
solution $(u_h,p_h)$ are analyzed.
Finally, a uniform $H^1$ and $L^2$-error estimates of optimal order for finite
element solution $(u_h,p_h)$ is obtained
if the uniqueness condition is satisfied.

DCDS-B

We carry out error estimation of a class of immersed finite element
(IFE) methods for elliptic interface problems with both perfect and
imperfect interface jump conditions. A key feature of these methods
is that their partitions can be independent of the location of the
interface. These quadratic IFE spaces reduce to the standard
quadratic finite element space when the interface is not in the
interior of any element. More importantly, we demonstrate that these
IFE spaces have the optimal (slightly lower order in one case)
approximation capability expected from a finite element space using
quadratic polynomials.

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