Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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<strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 65<br />
norm is the least-squares norm given by<br />
∫<br />
I = |F(ξ,t) − ξ| 2 dξ. (3.39)<br />
Ω C<br />
Minimizing I subject to the equidistribution principle is in fact the celebrated<br />
Monge–Kantorovich problem from differential geometry. This principle<br />
is often called optimum transport, as it leads to a transformation, the<br />
creation of which takes a minimum amount of work as a deviation from the<br />
identity. Intuitively, this is likely to deliver a regular mesh, as this mesh<br />
will be as close (in an averaged sense) to the most regular possible mesh,<br />
i.e., a completely uniform one. Remarkably, this minimization problem has<br />
a unique solution <strong>with</strong> a very elegant expression for the transformation.<br />
Theorem 3.3. There exists a unique optimal mapping F(ξ,t) satisfying<br />
the equidistribution equation. This map has the same regularity as M.<br />
Furthermore, F(ξ,t) is the unique mapping from this class which can be<br />
written as the gradient (<strong>with</strong> respect to ξ) ofaconvex (mesh) potential<br />
P (ξ,t), so that<br />
F(ξ,t)=∇ ξ P (ξ,t), ∆ ξ P (ξ,t) > 0. (3.40)<br />
Proof. See Brenier (1991) or Caffarelli (1992, 1996) for an abstract proof<br />
and Delzanno et al. (2008) for a proof in the context of adaptive mesh<br />
generation.<br />
The following is then immediate.<br />
Lemma 3.4. The map F is irrotational so that ∇ ξ × F = 0, and the<br />
Jacobian of F is symmetric.<br />
Significantly, the transformation above is an example of a Legendre transformation<br />
(Sewell 2002). Such transformations include translations and<br />
linear maps by positive definite symmetric matrices. We now show how to<br />
calculate such a transformation.<br />
3.3.2. Properties of optimally transported meshes<br />
It is immediate that if x = ∇ ξ P then<br />
∂X<br />
= H(P ),<br />
∂ξ<br />
where H(P ) is the Hessian of P . Additionally, if the measure M ∈ W 2 (Ω P )<br />
is strictly positive on its supports (assumed to be convex), then the potential<br />
P ∈ Wloc 2 (Ω C), and satisfies, in the classical sense, the Monge–Ampère<br />
equation<br />
M(∇ ξ P, t)H(P )=θ(t). (3.41)