Reap Sow and StepMonitor with FEM in Mathematica

Questions
1

Hello,

Can anyone help me figure out how to use Reap, Sow and StepMonitor with NDSolve applications using finite element module?

Thanks in advance.

Housam

KFUPM, KSA

2

Here’s how to do it with a one-dimensional problem.

In[1]:= res = 
  Reap[NDSolve[{y'[t] == y[t], y[0] == 1}, y, {t, 0, 1}, 
    EvaluationMonitor :> Sow[y[t]]]];

In[2]:= res

Out[2]= {{{y -> InterpolatingFunction[{{0., 1.}}, <>]}}, {{1., 
   1.00009, 1.00009, 1.00018, 1.00018, 1.00471, 1.00471, 1.00926, 
   1.00926, 1.01383, 1.01383, 1.03331, 1.03331, 1.02801, 1.02801, 
   1.0424, 1.0424, 1.05699, 1.05699, 1.07178, 1.07178, 1.10199, 
   1.10199, 1.13305, 1.13305, 1.16498, 1.16498, 1.19781, 1.19781, 
   1.23157, 1.23157, 1.31583, 1.31583, 1.40584, 1.40584, 1.50202, 
   1.50202, 1.60477, 1.60477, 1.71456, 1.71456, 1.83185, 1.83185, 
   2.02451, 2.02451, 2.23743, 2.23743, 2.46616, 2.46616, 2.71828, 
   2.71828}}}
3

Dear Frank,

I know this. Can you do something similar with NDSolve for PDEs using Mathematica’s finite element package?

Thanks much,

Housam

4

Doesn’t seem to work.

In[5]:= Needs["NDSolve`FEM`"]

In[8]:= \[CapitalOmega] = 
  ImplicitRegion[! ((x - 5)^2 + (y - 5)^2 <= 3^2), {{x, 0, 5}, {y, 0, 
     10}}];

In[11]:= op = -Laplacian[u[x, y], {x, y}] - 20;

In[9]:= Subscript[\[CapitalGamma], 
  D] = {DirichletCondition[u[x, y] == 0, x == 0 && 8 <= y <= 10],
   DirichletCondition[u[x, y] == 100, (x - 5)^2 + (y - 5)^2 == 3^2]};

In[13]:= res = 
 Reap @ NDSolveValue[{op == 
     0, {DirichletCondition[u[x, y] == 0, x == 0 && 8 <= y <= 10], 
     DirichletCondition[
      u[x, y] == 100, (-5 + x)^2 + (-5 + y)^2 == 9]}}, 
   u, {x, y} \[Element] 
    ImplicitRegion[(-5 + x)^2 + (-5 + y)^2 > 9 && 0 <= x <= 5 && 
      0 <= y <= 10, {x, y}], EvaluationMonitor :> Sow[{x, y, u[x, y]}]]

Out[13]= {InterpolatingFunction[{{0., 5.}, {0., 10.}}, <>], {}}