Adaptivity with moving grids

Adaptivity with moving grids 25
which M was determined in terms of apriorierror estimates (typically proportional
to the higher derivatives of the solution or estimates of these) was
given in Dorfi and Drury (1987), and is discussed in more detail presently.
More recently, monitor functions determined by a posteriori error estimates
have been constructed. An example of these, in the context of a piecewise
linear finite element approximation u h to a function u, isM = √ 1+αζ 2 ,
where
∑
∫
|u − u h | 2 1,Ω P
∼ ζ 2 (u h ) ≡
[∇u h .n l ] 2 l dl (2.20)
l: interior edge
and [.] l is the jump in the computed solution along the element edges. This
monitor function is used by Tang (2005) to compute solutions adaptively to
the Navier–Stokes equations with thin shear layers and/or high Mach numbers.
Similarly, in a series of papers studying both isotropic and anisotropic
meshes (Huang 2001a, 2001b, 2005a, 2005b, 2007), Huang explicitly considers
monitor functions which are designed to control the regularity, alignment
and quality of the mesh. These include monitor functions which are based
as estimates of the interpolation error of the computed solution, and we
consider them presently. Other measures of mesh quality can be incorporated
into the monitor function including maximum and minimum angle
conditions (Zlamal 1968, Babuška and Rheinboldt 1979), conditions on aspect
ratio and quantities that combine both shape and solution behaviour
(Berzins 1998). Finally, it is sometimes possible in the case of PDEs with
strong scaling structures (such as problems related to combustion and gas
dynamics) to find suitable monitor functions which give meshes reflecting
the natural scales of the problem (Budd and Williams 2006). We give an
example of these in Section 5, looking at a PDE which has solutions which
blow up in a finite time. In this case we need a fine mesh when the solution
is large, and take M(u) = √ a 2 + b 2 u 2p ,p>0.
We note at this stage that most choices of monitor function need a degree
of smoothing and regularization to perform effectively, and we will consider
this presently.
l
2.5.3. Matrix-valued monitor functions
The monitor function defined above is a scalar measure and is effective
in the specification and generation of certain isotropic meshes. However,
much greater freedom in mesh calculation may be required when calculating
anisotropic meshes, and in this case a matrix-valued monitor function
M can be used. In this case the meshes are defined via the metric determined
by an n × n matrix-valued monitor function that specifies the shape,
size and orientation of the elements throughout the physical domain Ω P .
Huang (2007) defines a matrix-valued monitor function M(x) using the