Adaptivity with moving grids

12 C. J. Budd, W. Huang and R. D. Russell
so that
Ω C =[0, 1] n .
To describe a computational mesh in the case of such simply connected
domains we typically divide Ω C ⊂ R n into N n uniform, regular tetrahedra or
cuboids of side proportional to 1/N and volume proportional to 1/N n ,and
we will initially assume that this is the case. In the r-adaptive procedure
considered in this section we consider the mesh points to be joined in a
simple (logically rectangular or triangular) network, the topology of which
(and consequently the ordering of the nodes in the network) is fixed for most
(if not all) of the time during the computation. Indeed, it is this constancy
of ordering which makes the r-adaptive procedure very attractive for finite
element and related computations.
To derive a moving mesh, the computational domain with its associated
mesh is then mapped to a physical domain Ω P ∈ R n , in which the underlying
PDE is posed. We assume that there is an invertible, adaptive mesh
generating function
F :Ω C → Ω P
describing this map, so that F is smooth on the interior of Ω C and continuous
on Ω C . Throughout this article we will denote variables in Ω C by Greek
letters, e.g., ξ, andinΩ P by Roman letters, x, and consider the function
F(ξ,t)tobetime-dependent. The action of the function F on the fixed
mesh τ C generates a moving mesh τ P in the physical domain. An example
of such a mesh is given in Figure 2.1, in which a uniform rectangular mesh
in Ω C is mapped to a mesh τ P . (This map was constructed by using the
optimal transport method described in Section 3.)
In the case where τ C is a uniform rectangular mesh, the resulting mesh
τ P in the physical space is then (in the representative example of a twodimensional
system) given by the points (X i,j ,Y i,j ), where F =(x, y) and
( i
X i,j = x
N , j ) ( i
, Y i,j = y
N N , j )
. (2.1)
N
We assume further that the boundary of ∂Ω C of Ω C is mapped by F to
the boundary of ∂Ω P of Ω P . In some r-adaptive strategies (such as the
multi-equidistribution and/or variational strategies described in Huang and
Russell (1999), the map F is augmented with a second map,
∂F : ∂Ω C → Ω P ,
which explicitly describes the map from one boundary to another. This
has the advantage of close control of the meshing strategy right up to the
boundary, but has the disadvantage of introducing extra complexity into
the system. In other algorithms, such as the optimal mapping strategy
(Delzanno, Chacón, Finn, Chung and Lapenta 2008, Budd and Williams