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In the first study, we consider problems with interfaces,
where the unknown or its derivatives may
have jump discontinuities. Finite difference methods,
including the method of A. Mayo and the immersed interface
method of R. LeVeque and Z. Li, maintain accuracy by adding
corrections, found from the jumps, to the difference operator at
grid points near the interface and modifying the operator
if necessary. It has long been observed that the solution
can be computed with uniform O(h^2) accuracy even if
the truncation error is O(h) at the interface, while
O(h^2) in the interior. We prove this fact for a
class of static interface problems of elliptic type
using discrete analogues
of estimates for elliptic equations. Moreover, we show
that the gradient is uniformly accurate to O(h^2 log{(1/h)}).
Various implications are discussed, including the accuracy
of these methods for steady fluid flow governed by the
Stokes equations. Two-fluid problems can be handled by
first solving an integral equation for an unknown jump.
Numerical examples are presented which
confirm the analytical conclusions, although the observed error in the gradient
is O(h^2).
Model configuration for water-permeable boundary
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In another study, we present
a mathematical model to simulate water and solute transport in a
highly viscous fluid with a water-permeable, elastic immersed membrane.
In this model, fluid motion is described by Stokes flow, whereas
water fluxes across the membrane are driven by transmural pressure and
solute concentration differences.
The elastic forces, arising from the membrane being distorted from
its relaxed configuration, and the transmembrane water fluxes introduce
into model solutions discontinuities across the membrane.
Such discontinuities are faithfully captured using a second-order
explicit jump method, in which jumps in the solution
and its derivatives are incorporated into a finite-difference scheme.
Numerical results suggest that the method exhibits desirable volume accuracy
and mass conservation.
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