Solving PDE with Dirichlet, Neumann and Boundary conditions

Question

I am trying to solve the following PDE: $$ u_{xx} + u_{yy} = \begin{cases} - \cos(x) \quad -\pi/2 \leq x \leq \pi/2, \\ 0 \quad \text{otherwise} \end{cases} $$

The domain is $\Omega = [-\pi,\pi] \times [0,2]$. The boundary conditions are periodic at $x = \pm \pi$ - $u(-\pi,y) = u(\pi,y)$ - and $u(x,0) = 0$ (Dirchlet) and $u_y(x,2) = 0$ (Neumann).

I am able to solve the PDE if I choose all of conditions to be Dirichlet. I cannot, however, solve the PDE in the case where I have a combination of Neumann, Dirichlet, Periodic boundary conditions. I have no idea how to implement the Neumann boundary condition appropriately. Also my implementation of periodic boundary conditions gives errors.

Ω = Rectangle[{-Pi, 0}, {Pi, 2}];
op = Laplacian[u[x, y], {x, y}] - 
   Piecewise[{{-Cos[x], -Pi/2 < x < Pi/2}, {0, x < -Pi/2}, {0, 
      x > Pi/2}}, {x, -Pi, Pi}];
Subscript[Γ, 
   D] = {DirichletCondition[u[x, y] == 0, y == 0], 
   DirichletCondition[u[x, y] == 0, y == 2 ]};
mapping = 
  Last[FindGeometricTransform[{{-Pi, 0}, {-Pi, 2}}, {{Pi, 0}, {Pi, 
      2}}]];
Subscript[Γ, P] = 
  PeriodicBoundaryCondition[u[x, y], x == -Pi, mapping];
ufun = NDSolveValue[{op == 0, Subscript[Γ, D], 
    Subscript[Γ, P]}, 
   u, {x, y} ∈ Ω];
ContourPlot[ufun[x, y], {x, y} ∈ Ω, 
  ColorFunction -> "Temperature", AspectRatio -> Automatic];

Answer 1

4 issues here, 2 are simple mistakes, 2 are not.

1. Though it happens not to cause problem in this case, your understanding for syntax of Piecewise is wrong, please check its document carefully. The correct one should be Piecewise[{{-Cos[x], -Pi/2 < x < Pi/2}}].

2. You've inverted the direction in FindGeometricTransform.

3. Currently PeriodicBoundaryCondition and DirichletCondition cannot overlap even at points, personally I think this design is a bit unnecessary and inconvenient.

4. There's an issue related to periodic b.c. of FiniteElement method, which is discussed in this post. To obtain the desired solution, we need to set 2 PeriodicBoundaryConditions and a ghost layer.

To sum up:

eps = 0.1;
Ω = Rectangle[{-Pi, 0}, {Pi + eps, 2}];
offset = 1.5;
op = Laplacian[u[x, y], {x, y}] - 
   Piecewise[{{-Cos[x], -Pi/2 + offset < x < Pi/2 + offset}}];
Subscript[Γ, D] = 
  DirichletCondition[u[x, y] == 0, y == 0];
mapping = Last[FindGeometricTransform[{{Pi, 0}, {Pi, 2}}, {{-Pi, 0}, {-Pi, 2}}]];
mappinginverse = 
  Last[FindGeometricTransform[{{-Pi, 0}, {-Pi, 2}}, {{Pi, 0}, {Pi, 2}}]];
Subscript[Γ, 
   P] = {PeriodicBoundaryCondition[u[x, y], x == -Pi && 0 < y < 2, mapping], 
   PeriodicBoundaryCondition[u[x, y], x == Pi + eps && 0 < y < 2, mappinginverse]};
ufun = NDSolveValue[{op == 0, Subscript[Γ, D], 
    Subscript[Γ, P]}, u, {x, y} ∈ Ω];
ufunwrong = 
  NDSolveValue[{op == 0, Subscript[Γ, D], 
    Last@Subscript[Γ, P]}, u, {x, y} ∈ Ω];
With[{y = 1.5}, 
 Plot[{ufun[Mod[x, 2 Pi, -Pi], y], ufunwrong[Mod[x, 2 Pi, -Pi], y]} // 
   Evaluate, {x, -Pi, 3 Pi}, PlotStyle -> {Automatic, Dashed}]]

Notice I've set offset = 1.5 to illustrate the necessity of double periodic b.c., for your original problem, set offset = 0.

"Wait, where's the Neumann b.c.?" Your Neumann b.c. corresponds to a zero NeumannValue. As mentioned in Details section of document of NeumannValue:

...not specifying a boundary condition at all is equivalent to specifying a Neumann 0 condition.

So you don't need to write code for your Neumann b.c..

Answer 2

modified

Thanks to @xzczd for the helpful discussion

Here my simplified version without ghost layer and mapping:

The PeriodicBoundaryCondition's have to be defined on both sides x==-Pi and x==Pi. y==0 must be excluded, otherwise there is a conflict with DirichletCondition.

Using a triangle-mesh NDSolve is able to correctly handle the periodic bc's

\[CapitalOmega] = Rectangle[{-Pi, 0}, {Pi , 2}];
offset = 1.5;
pde = Laplacian[u[x, y], {x,y }] == -Cos[x] Boole[-Pi/2 + offset < x< Pi/2 + offset];
\[CapitalGamma]D = DirichletCondition[u[x, y] == 0, y == 0];
[CapitalGamma]P = {
PeriodicBoundaryCondition[u[x, y],x ==  [Pi] && 0 < y <= 2, TranslationTransform[{-2 Pi, 0}]], 
PeriodicBoundaryCondition[u[x, y], x == -[Pi] && 0 < y <= 2, TranslationTransform[{2 Pi, 0}]]}
ufun = NDSolveValue[{pde, [CapitalGamma]D, [CapitalGamma]P},u, {x, y} [Element] [CapitalOmega], 
Method -> {"FiniteElement","MeshOptions" -> {"MeshElementType" ->"TriangleElement",    "MeshOrder" -> 1}}]
Plot3D[ufun[x, y], Element[{x, y}, [CapitalOmega]],AxesLabel -> {x, y, u[x, y]}, MeshFunctions -> {#2 &}]

The periodic boundary conditions are well fullfilled now

Plot [Evaluate[Table[ufun[x, y], {y, Subdivide[0, 2, 10]}]], {x, -Pi,Pi}]

The solution QP asked for follows with offset==0