Downsampling (interpolating from refined to coarse) in Hermite spaces

[nadav7679]

Hi all,

I computed a dataset of functions on a Hermite FE space built on PeriodicIntervalMesh with 4096 cells.
Now I want to replicate that dataset on a few, more coarse, grids (ncells= 2048, 1024, 512, ...), all in Hermite FE space.

If it were a CG space I would simply interpolate, but from what I understand, Firedrake doesn't currently support interpolating into Hermite spaces.

Does anyone have an idea on how to duplicate the dataset on coarse spaces?

Many thanks!

[colinjcotter]

If you are coarsening anyway, might as well interpolate to CG1 first, then transfer to the other mesh. Then project into H3 if you need to.

[colinjcotter]

You can project into any space.

[nadav7679]

So a sequence of actions like this (given mesh2 the coarse one) :

her1 = FunctionSpace(mesh1, "HER", 3)
her2 = FunctionSpace(mesh2, "HER", 3) 
cg1 = FunctionSpace(mesh1, "CG", 1)
cg2 = FunctionSpace(mesh2, "CG", 1)

func_her1 = Function(her1, val=dataset[index, :])
func_her2 = Function(her2)
func_cg2 = Function(cg2)
func_cg1 = Function(cg1)

func_cg1.interpolate(func_her1)
func_cg2.interpolate(func_cg1)
func_her2.project(func_cg2)

Can we account for the loss of accuracy in this case?

[nadav7679]

If anyone is interested, the above code plus this snippet,

func_cg2_projected = Function(cg2)
func_cg2_projected.interpolate(func_her2)

fig, axes = plt.subplots()
plot(func_cg1, axes=axes, label="HER1 -> interpolate -> CG1")
plot(func_cg2, axes=axes, label="CG1 -> interpolate -> CG2")
plot(func_cg2_projected, axes=axes, label="CG2 -> project -> HER2 -> interpolate -> CG2")
axes.legend()

produced this image that shows the differences between the different operations:

While there is a difference between the CG2 interpolation and the HER2 projection (orange and green), I believe it is small enough for me to use this method.
Thanks!

[pbrubeck]

I think you want HER(fine)->CG3(fine)->CG3(coarse)->HER(coarse)

[nadav7679]

I think you want HER(fine)->CG3(fine)->CG3(coarse)->HER(coarse)

It does make more sense, but I get a strange behaviour when interpolating HER(fine) into CG3(fine):
(note that in the following code snippet I use only the original (refined) mesh and the numbers in the variables mean something different than the above snippets)

her = FunctionSpace(mesh1, "HER", 3)
cg3 = FunctionSpace(mesh1, "CG", 3)
cg1 = FunctionSpace(mesh1, "CG", 1)

func_her = Function(her1, val=dataset[index, :])
func_cg1 = Function(cg1)
func_cg3 = Function(cg3)

func_cg3.interpolate(func_her)
func_cg1.interpolate(func_her)

fig, axes = plt.subplots()

plot(func_cg1, axes=axes, label="CG1")
plot(func_cg3, axes=axes, label="CG3")
axes.legend()

produced: