4.23 How to solve Poisson PDE in 2D with Neumann boundary conditions using Finite Elements

Solve \(\bigtriangledown ^2 T -20 T =-20\) on the unit square \(0\leq x \leq 1,0 \leq y \leq 1\) subject to insulated boundary conditions at \(x=0\) and \(y=0\) and \(T=0\) at \(x=1,y=1\).

4.23.1 Mathematica

In Mathematica 10.0, one can use Finite elements directly. Here is the code to solve the above.

Clear[u,x,y]; 
r=NDSolveValue[{D[u[x,y],{x,2}]+D[u[x,y],{y,2}]-20*u[x,y]== 
    -200+NeumannValue[0,x==0] 
    +NeumannValue[0,y==0], 
    DirichletCondition[u[x,y]==0,x==1], 
    DirichletCondition[u[x,y]==0,y==1]}, 
    u,{x,0,1},{y,0,1}, 
    Method->{"FiniteElement", 
      "MeshOptions"->{"BoundaryMeshGenerator"->"Continuation"}}];
 

Plotting the solution

Plot3D[r[x,y],{x,0,1},{y,0,1},Mesh->All]
 

Now we make a contour plot

ContourPlot[r[x,y],{x,0,1},{y,0,1}]