Adaptivity with moving grids

Adaptivity with moving grids 27
conditions if it is to be applied in dimension n>1. The determination of
these additional conditions is neither straightforward nor unique, and leads
to a variety of different methods for mesh generation, for which control of
mesh skewness and other geometric properties must also be considered. Two
examples of the additional conditions might be to impose an irrationality
condition in the computational domain so that ∇ ξ × F = 0 (Budd and
Williams 2006) – which is the basis of the optimal transport methods – or
to require that the mesh velocity is irrotational in the physical domain so
that ∇ x × v =0(Caoet al. 2002) – which is the basis of the GCL methods.
The augmented equations can then be solved in a number of ways to find the
mesh: by directly solving the nonlinear system, which can be expensive; by
differentiating the condition and solving the resulting differential equations
which leads to the GCL methods we will consider in Section 4; by relaxing
towards a solution of the system, which leads to the MMPDE methods in one
and higher dimensions; or to have a global variational principle associated
with the error and to find the gradient flow equations associated with it.
We consider the latter now, with more details in Section 3.
2.6.2. Variational methods in one dimension and links to equidistribution
An alternative strategy for determining a mesh, also based on an appropriate
monitor function, is the variational method. In such a method the
stationary points determine the optimal mesh, and the associated gradient
flow equations towards the stationary points determine a suitable mesh
motion strategy.
Suppose that I(ξ) is a certain functional and that the mesh generation
strategy is equivalent to minimizing I over a certain function space. Finding
the Euler–Lagrange equations then leads to a gradient flow equation to
evolve the mesh towards the equilibrium state (a stationary point of I),
which is given by
∂ξ
∂t = −δI δξ . (2.25)
This can then lead directly to an MMPDE to move the mesh by introducing
some additional local control on the mesh movement in the form
∂ξ
∂t = −P τ
δI
δξ , (2.26)
where P is a positive differential operator and τ > 0 is a parameter for
adjusting the time scale of the mesh movement. In one dimension, equidistributing
the scalar monitor function M is exactly equivalent to minimizing
the functional
I(ξ) = 1 2
∫ 1
0
1
M
( ) ∂ξ 2
dx. (2.27)
∂x