Adaptivity with moving grids

38 C. J. Budd, W. Huang and R. D. Russell
It is interesting to point out that uniform convergence has been obtained
by a number of researchers for equidistributing meshes determined apriori
by the exact solution or the singularity information of the exact solution
for singularly perturbed differential equations: e.g., see Sloan et al. (Qiu
and Sloan 1999, Qiu, Sloan and Tang 2000), Mackenzie et al. (Mackenzie
1999, Beckett and Mackenzie 2000, Beckett, Mackenzie, Ramage and Sloan
2001b, Beckett and Mackenzie 2001a, 2001b, Mackenzie and Mekwi 2007b),
and Huang (2005c). Convergence results are also obtained for a posteriori
equidistributing meshes determined by computational solutions for differential
equations in Babuška and Rheinboldt (1979), Kopteva and Stynes
(2001), He and Huang (2009) and Huang, Kamenski and Lang (2009).
2.8.2. Static interpolation error
In higher dimensions it is very hard to obtain reliable estimates for the truncation
error when solving a general PDE. A somewhat easier, but still very
important, question to address is whether a mesh is suitable to approximate
the solution of the PDE, in particular to interpolate the solution. We now
consider this question.
Suppose that the solution of the differential equation (or indeed any appropriate
function defined in the physical domain) is given by u(x,y,...).
For the case of a problem in two dimensions we can define the point values
of u on the non-uniform mesh by
U i,j = u ( X i,j ,Y i,j
)
.
A natural measure of error is the interpolation error obtained by approximating
u on the mesh with suitable functions using the above point values.
Significant progress in finding meshes with good properties in reducing
the interpolation error of a solution has been made in this direction in
the past decade. Formulae giving the optimal monitor function to minimize
this error over a suitable mesh have been developed based on interpolation
error estimates by Huang and Sun (2003) in the H m -norm, Chen
et al. (2007) in the L q -norm, Huang (2005a, 2005b) intheW m,q -norm,
and Cao (2005, 2007a, 2007b, 2008) for higher-order interpolation in two
dimensions. Formulae have also been developed based on an a posteriori
error estimate for one dimension (He and Huang 2009), a hierarchical basis
a posteriori error estimate (Huang et al. 2009), and semi a posteriori error
estimates for variational problems (Huang and Li 2009). Formulae for
the monitor functions in these cases can be obtained as follows (Huang and
Sun 2003, Huang 2005a). A so-called anisotropic error bound, taking into
consideration the directional effect of the error or solution derivatives, is first
developed. This error bound can be regarded as a function of the monitor
function M when only meshes satisfying the alignment and equidistribution