Adaptivity with moving grids

44 C. J. Budd, W. Huang and R. D. Russell
Consequently, when we make use of the regularized monitor function ˆM to
define the mesh we have
∫
|A ′ A
| =
ˆM
∫
dx
A
≈
M dx ≈ 1 2θ 2θ 2 ,
and
Λ ≈ 2|Ω P |
− 1 ≈ 1.
|B|
This gives the desired 50:50 quality to the mesh.
3. Location-based moving mesh methods
In this section we will look in greater detail at the various methods described
in Section 2 under the general heading of location-based methods. These are
those methods which determine the location (or more precisely the density)
of the mesh points, typically through solving some form of nonlinear differential
equation through some form of gradient flow method. The latter
can be hard to solve. However, the advantage of these methods is that they
tend to give meshes with good global properties, avoiding excessive skewness.
We will consider in detail the various methods outlined in the previous
section, such as MMPDE-based methods, variational methods and optimal
transport methods.
3.1. MMPDE methods in one dimension
Methods based on moving mesh partial differential equations (MMPDEs)
are now universally used as a means of r-adaptivity in one dimension, and
have been incorporated into codes such as MOVCOL and AUTO. There are
many different MMPDEs, which together encapsulate most of the methods
used to derive adaptive meshes in one dimension.
We consider a one-dimensional map x(ξ,t) from[0, 1] to [a, b] with associated
mesh points X i = x(i∆ξ,t), which equidistributes the monitor function
M. This map satisfies the equidistribution equation
Mx ξ = θ, x(0,t)=a, x(1,t)=b, θ =
∫ b
a
M dx. (3.1)
Lemma 3.1. The equidistribution equation (3.1) has a unique monotone
increasing solution x(ξ,t) for all M>0.
Proof. Integrating (3.1) with respect to ξ and changing variables gives
∫ x
∫ Xi
M dx ′ = θξ or M dx = 1 M dx. (3.2)
a
X i−1
N a
Now, as M>0 the left-hand side of this expression is a monotone increasing
∫ b