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