Gradients of the NDSolve PDE solution

Question

I would like to understand, how to obtain gradients of the PDE solution obtained with NDSolve. To be precise let us consider a Laplace equation from one of the examples:

    Clear[x, y, f];
Needs["NDSolve`FEM`"]
emesh = ToElementMesh[Disk[]];

f = NDSolveValue[{Derivative[0, 2][u][x, y] + 
Derivative[2, 0][u][x, y] == 0, 
   DirichletCondition[u[x, y] == Sin[x y], True]}, 
  u, {x, y} ∈ emesh]

This returns the interpolation function which one can plot and integrate:

    NIntegrate[f[x, y], {x, y} ∈ emesh]

(*  1.52794*10^-8  *)

Plot3D[f[x, y], {x, y} ∈ emesh]

enter image description here

This, however, does not work:

  g[x_, y_] := D[f[x, y], x];
Plot3D[g[x, y], {x, y} ∈ emesh]

Since Integrate does not work on this result, only NIntegratedoes, the problem is probably that one needs to apply a numeric derivative. What and how?

@MichaelE2 but this is completely different: its on a mesh :-) — chris, Jun 08 '15 at 16:05
@chris I'm a little slow to understand the ":-)" -- I thought you would object that it's a two-variable function (instead of single-variable). :-) — Michael E2, Jun 08 '15 at 16:14

chris · Accepted Answer · 2015-06-08T15:20:28.737

3

That's an easy one:

g[x_, y_] = D[f[x, y], x];
Plot3D[g[x, y], {x, y} \[Element] emesh]

Mathematica graphics

Note the : in the definition of g :-)

edited Jun 08 '15 at 15:20

answered Jun 08 '15 at 15:19

chris

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You beat me to it by a few seconds :-) – user21 Jun 08 '15 at 15:20
@user21 I think I learnt it from one of your posts! – chris Jun 08 '15 at 15:21
1

If one wanted it to be SetDelayed for some reason one could also do g[x_, y_] := Evaluate[D[f[x, y], x]]; – user21 Jun 08 '15 at 15:23
@user21 feel free to answer this question instead http://mathematica.stackexchange.com/q/84726/1089 :-) it is more in your league ! – chris Jun 08 '15 at 15:23
I am very much tempted but I'd like to wait a bit... ;-) – user21 Jun 08 '15 at 15:30
2

One can also use g = Derivative[1, 0][f]. – Michael E2 Jun 08 '15 at 16:09

Gradients of the NDSolve PDE solution

1 Answers1