Adaptivity with moving grids

18 C. J. Budd, W. Huang and R. D. Russell
it is in general very hard to do this, and instead some property of J (such as
its determinant) is prescribed, and this is then used to determine the mesh.
A mesh can also be made smoother by some direct methods. For example,
(weighted) Laplace smoothing is often used in hp adaptation (see Carey
(1997)). In this strategy, the coordinates of an interior mesh point are
adjusted so that they become the (weighted) average of the coordinates
of its neighbouring points. Typically this is carried out in a Jacobian or
Gauss–Seidel fashion. When this is the case, Laplace smoothing can be
viewed as the application of the Jacobian or Gauss–Seidel iteration to the
solution of a discretization of the partial differential equation
−∆ ξ (ˆx, ŷ) =(x, y), (2.7)
where ∆ ξ is the Laplacian operator applied in the computational domain.
In r-adaptive methods based on equidistribution, on the other hand, a
smoother mesh is often obtained indirectly by smoothing the monitor function
M used for controlling mesh adaptation and movement. We will describe
this strategy presently.
When calculating solutions to a (partial) differential equation on a nonuniform
mesh it is essential that there is a strong control on the mesh
variation. For one-dimensional meshes for which we have a mesh function
x(ξ), mesh points X i = x(i∆ξ) and local mesh spacing given by ∆ i =
X i+1 − X i then the grid size ratio or local stretching factor r is given by
r =
∆ i
. (2.8)
∆ i−1
In a uniform mesh we have that r = 1. For many calculations on a nonuniform
mesh, we require instead that
r =1+O(∆ i ). (2.9)
Such grids are termed quasi-uniform (Li et al. 1998, Zegeling 2007, Kautsky
and Nichols 1980, Kautsky and Nichols 1982), and normally lead to
truncation (and approximation) errors of the same order as uniform meshes
(Veldman and Rinzema 1992). We note that
r =1+ ∆ i − ∆ i−1
∆ i
Consequently, as ∆ i ≈ ∆ξx ξ , etc., wehave
=1+ X i+1 − 2X i + X i−1
X i − X i−1
.
r =1+∆ i
x ξξ
x 2 ξ
+ O(∆ 2 i ).
Thus the mesh is quasi-uniform provided that
Λ ≡ x ξξ
x 2 = O(1). (2.10)
ξ