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