Adaptivity with moving grids

Adaptivity with moving grids 63
Example 3. This example generates an adaptive mesh for a given analytical
solution
( (
u(x, y) = tanh 30 x 2 + y 2 − 1 ))
8
( (
+tanh 30 (x − 0.5) 2 +(y − 0.5) 2 − 1 ))
8
( (
+tanh 30 (x − 0.5) 2 +(y +0.5) 2 − 1 ))
8
( (
+tanh 30 (x +0.5) 2 +(y − 0.5) 2 − 1 ))
8
( (
+tanh 30 (x +0.5) 2 +(y +0.5) 2 − 1 ))
8
defined in [−2, 2] × [−2, 2]. An adaptive mesh with a monitor function
is based on isotropic and anisotropic estimates error in interpolating this
function. This is expected to concentrate around five circles. Results are
shown in Figure 3.5.
Example 4.
for
This example generates a three-dimensional adaptive mesh
u(x, y, z) = tanh(100((x − 0.5) 2 +(y − 0.5) 2 +(z − 0.5) 2 ) − 0.0625)
defined in the unit cube. An adaptive mesh with a monitor function is based
on the error in interpolating this function. This is expected to concentrate
near the sphere centred at (0, 0, 0) with radius 0.25. Results are shown in
Figure 3.6.
3.3. Optimal transport methods
3.3.1. Derivation of the optimal transport equations
Optimal transport methods are a very natural generalization of MMPDE
methods in one dimension, that retain much of the simplicity of the onedimensional
approach (such as always solving scalar equations and automatic
calculation of the mesh on a boundary) whilst being general enough
to deliver meshes of provable mesh quality (with many of the proofs following
directly from the one-dimensional case). They have the disadvantage of
being less flexible than some of the moving mesh methods described above.
However, in practice they can give very regular meshes for a wide range of
possible monitor functions. The key idea behind an optimal mesh is that
it should be one which is closest to a uniform mesh in a suitable norm,
consistent with satisfying the equidistribution principle. The simplest such