Adaptivity with moving grids

86 C. J. Budd, W. Huang and R. D. Russell
the computational space. Unlike the MFE methods, this partial differential
equation is well-defined for all time. In a similar manner to the optimal
mesh methods, we need only solve a scalar equation; however, unlike the
optimal mesh methods this equation is linear in the physical coordinates.
Observe, further, that unlike certain of the MMPDE-based and variationalbased
methods described in Section 3, this system automatically deals with
the mesh location on the boundaries. Having determined φ from this equation,
the mesh velocity v can be determined directly, and the mesh then
found from integrating the equation x t = v with respect to time using, for
example, an SDIRK method. This time integration to give x has the disadvantage
of possibly introducing mesh tangling, and of mesh points moving
out of the domain during the course of the integration, if an appropriately
coarse discretization is used. This method is described in Cao et al. (2002)
An alternative formulation, also described in Cao et al. (2002), considers
a variational formulation, where it is shown that the solution of (4.3) is also
the minimizer of the functional
I[v] = 1 2
∫Ω P
(
|∇ · (Mv)+M t | 2 +
( M
w
) 2
|∇ × w(v − u)| 2 )
dx. (4.7)
This equation can be discretized using a finite element method with basis
functions defined over the mesh in the physical space, and v found using
a simple Galerkin method. The mesh points can then also be found from
v through a Galerkin calculation. Details of these calculations are given in
Baines et al. (2005, 2006), where the GCL method is coupled to an ALE (arbitrary
Lagrangian–Eulerian) method for discretizing the underlying PDE.
In the usual implementation of the GCL method the weight function
w =1
is taken. This gives an irrotational mesh (in the physical coordinates) when
the background velocity u = 0. In many implementations of GCL, u =0.
However, for certain applications, such as in computational fluid dynamics,
the background velocity can be taken to be the flow velocity.
The appeal of the GCL methods is that to find φ (and hence v) we need
only solve a scalar linear elliptic partial differential equation. This seems
to have the advantage over the optimal transport methods, which require
the solution of a nonlinear equation. However, like other velocity-based
methods it has the potential disadvantages of having problems with mesh
tangling and mesh skewness as the mesh points follow the moving features
in a monitor function. Furthermore, we require the solution of the mesh
equations in the physical domain rather than the computational domain.
This loses some of the speed advantages gained (by, for example, the use of
spectral methods) when solving the mesh equations on a very uniform mesh
in the computational domain.