Adaptivity with moving grids

Adaptivity with moving grids 55
smooth to avoid possible deterioration of accuracy: see, e.g., Mulholland,
Huang and Sloan (1998), Wang and Shen (2005), Feng, Yu, Hu, Liu, Du
and Chen (2006) and Tee and Trefethen (2006). While preliminary results
are promising, much further investigation is needed to determine the full
potential of these adaptive spectral methods.
3.2. MMPDEs and variational methods in n dimensions
3.2.1. Description of some variational-based methods
In the variational and MMPDE approaches of mesh adaptation in n dimensions,
briefly described in Section 2, adaptive meshes are also generated
as images of a computational mesh under a coordinate transformation from
the computational domain to the physical domain. Such a coordinate transformation
is determined by an adaptation functional, which is commonly
designed to measure the difficulty in the numerical approximation of the
physical solution. The functional often involves mesh properties and employs
a monitor function to control mesh quality and mesh concentration.
The key to the development of variational and MMPDE methods is the formulation
of the adaptation functional. Direct-use standard-error estimates
are often not appropriate since they often lead to non-convex functionals
in two and higher dimensions. Instead, most of the existing methods have
been developed based on physical, geometric, mesh quality control, and/or
other considerations.
The functional can be formulated in terms of either the coordinate transformation
x = F(ξ,t) or its inverse transformation ξ = F −1 (x,t). The latter
has been used more commonly than the former because it is less likely to
produce mesh tangling for non-convex domains (e.g., see Dvinsky (1991)).
In the latter case, the adaptation functional takes the general form
∫
I(ξ) = G(M, ξ, ∇ξ i )dx, i =1,...,n, (3.26)
Ω P
where G is a continuous function of its arguments, M is the (scalar or
matrix-valued) monitor function, and ∇ is the gradient operator with respect
to the physical coordinate x. Once a functional has been defined,
an MMPDE can be obtained as described in (2.26) as the gradient flow
equation of the functional (Huang and Russell 1997b, 1999), i.e.,
∂ξ i
∂t = −P ∂I
, i =1,...,n, (3.27)
ɛ ∂ξ i
where P is a positive differential operator and ɛ>0 is a parameter for
adjusting the time scale of mesh movement. For the general form (3.26),
this becomes
∂ξ i
∂t = P (
∇ ·
ɛ
∂G
∂(∇ξ i ) − ∂G
∂ξ i
)
, i =1,...,n. (3.28)