Adaptivity with moving grids

that
Adaptivity with moving grids 23
∫
ν(B) =
B
M(x) dx,
for any Borel subset B of Ω P , where dx is the usual Lebesgue measure on
Ω P . Furthermore M is unique up to sets of Lebesgue measure zero.
Proof. See Capiński and Kopp (2004).
The Radon–Nikodym theorem shows that for any invertible map F we
can find a unique function M(x) such that, for any set A ⊂ Ω C ,wehave
∫ ∫
dx = M(x)dx. (2.15)
A
F (A)
Note, however, that (other than the special case of one dimension) the
same function M may be associated with many different maps.
The function M(x) > 0 is a function of x and t, but is more usually
defined in terms of the solution u(x,t) of the underlying PDE, so that we
might have
M(x,t) ≡ M(x,u(x,t), ∇u(x,t),...,t).
In this context M is usually called a scalar monitor function, and is chosen
to be large when the mesh points need to be clustered, for example if the
solution of the underlying problem has a high gradient. In this case the
Lebesgue measure of the set B may be small even if the measure ν(B)
is not. This may occur, for example, in the neighbourhood of a solution
singularity or a sharp front. An obvious example of a monitor function is
some estimate of the truncation error in the calculation of the solution of the
underlying PDE, and this was the original motivation of the equidistribution
approach of de Boor (1973). Loosely speaking, equidistributing the error
in calculating the solution of a PDE over all mesh elements is a necessary
condition for finding a global minimum of that error (Johnson 1987)
Assume now that we know the scalar function M and consider how we
might determine an appropriate map F. To do this we introduce an arbitrary
non-empty set A ⊂ Ω C in the computational domain, with a corresponding
image set F(A,t) ⊂ Ω P . The map F equidistributes the respective
scalar monitor function M if the Stieltjes measures of A and F(A,t), normalized
over the measure of their respective domains, are the same. This
implies that
∫
∫ A dξ
∫
dξ = F(A,t) M(x,t)dx
∫Ω P
M(x,t)dX . (2.16)
Ω C