Adaptivity with moving grids

Adaptivity with moving grids 67
consider the bottom boundary of the square in the physical domain, for
which Y = 0. A grid line Γ which intersects this side is given by
Γ={(x, y) :0≤ η ≤ 1, ξ fixed}.
The tangent to this line in the physical domain is given by τ =(x η ,y η ) T ,
and, trivially, the tangent to the bottom boundary of the square by the vector
t =(1, 0) T . However, on the bottom boundary, the function P satisfies
the identity P η (ξ,0) = 0 and by definition x = P ξ . It follows immediately
that when η =0,
x η (ξ,0) = P ηξ (ξ,0) = 0,
so that on the lower boundary τ · t = 0. Thus the mesh is orthogonal
to the lower boundary. Similar results apply to the other sides as well.
The condition of mesh orthogonality at the boundary of the square is often
desirable in certain circumstances (Thompson et al. 1985). However, it
may cause problems when resolving features, such as fronts, which intersect
boundaries at an angle. In such cases it may be useful to refine the mesh
close to the boundary, either by introducing a finer computational mesh
there, or by locally increasing the value of the monitor function M at mesh
points adjacent to the boundary.
Mesh symmetries. Some very desirable properties of the mesh follow immediately
from the properties of the Monge–Ampère equation. It is trivial
to see that the equation is invariant under translations in (ξ,η). It is
also easy to see that (in a similar manner to the Laplacian operator) the
Monge–Ampère equation is also invariant under any orthogonal map such
as a rotation or a reflection This is because the Monge–Ampère equation
is the determinant of the Hessian, which transforms covariantly under such
maps. This simple observation implies that (away from boundaries) the
meshes generated by solving the Monge–Ampère equation should have no
difficulty aligning themselves to structures, such as shocks, which may occur
anywhere in a domain and at any orientation. It is immediate that the
Monge–Ampère equation is also invariant under scaling transformations of
the form ξ → λξ, η → µη, P → νP, provided that M is chosen carefully.
Mesh skewness. The regularity of the mesh generated by this approach gives
a useful feature in seeing control in the variation of the element size across
the domain. In general the meshes generated by the Monge–Ampère equation
have good regularity properties and are effective in interpolating functions
(Delzanno et al. 2008). It is possible to make some estimates for the
resulting skewness of the mesh in terms of the properties of the function P .
Consider a two-dimensional problem for which the map from Ω C to Ω P has