Adaptivity with moving grids

Adaptivity with moving grids 37
a true estimate, this expression needs to be integrated. A useful expression
for the error for both (2.46) and (2.45) (see Andreev and Kopteva (1998))
is then given by the next lemma.
Lemma 2.5.
[
]
‖U j − u(X j )‖ ∞ ≤ C max ∆ 2 i max
[X i ,X i+1 ] |u′′ | +∆ 2 i . (2.55)
Given suitable aprioriestimates, this error can be bounded by using a
monitor function M which controls this via the expression
(
) (
)
∆ 2 i max
[X i ,X i+1 ] |u′′ | +1 =∆ξ 2 x 2 ξ max
[X i ,X i+1 ] |u′′ | +1
=∆ξ 2( )
max
[X i ,X i+1 ] |u′′ | +1 θ 2 /M 2 .
This motivates the choice of the curvature-dependent monitor function given
by
M = √ 1+|u ′′ | 2 .
Such a function has been used by Blom and Verwer (1989); see also Mackenzie
and Robertson (2002), Chen (1994), Huang and Sun (2003) and Kopteva
(2007). In this case the error becomes a function of ∆ξ only and does not
depend upon the solution. Hence it has the great advantage of yielding a
mesh for which large variations in u ′′ (for example at boundary layers) do
not affect accuracy. This is good enough for most calculations. Observe,
however, the difference between using the curvature-based monitor function
to bound the error, and the optimal monitor function for the Poisson
equation which eliminates this error to leading order.
Similar issues arise in the case of the singularly perturbed problems (2.45).
For example, it is possible to get very sharp estimates on the solution in
certain cases (such as when c(x) = 1). In the latter case (Andreev and
Kopteva 1998) we have the following result.
Lemma 2.6.
‖U j − u(X j )‖ ∞ ≤ C [ ‖ min{∆ 2 i /ɛ 2 , 1} e −xi−1/ɛ ‖ ∞ + max ∆ 2 ]
i . (2.56)
This error can be completely controlled to be proportional only to ∆ξ 2
using a Bakhvalov mesh. For such problems it can also be shown (Kopteva
2007) that, if the monitor function is chosen to be a discrete arclength of
the form
M = √ 1+u 2 x,
then, provided that the solution has converged closely to an equidistributed
one, the computed solution is first-order accurate with errors O(∆ξ), independent
of the value of ɛ.