Solve a Poisson Equation with Periodic Boundary Conditions

Solve a Poisson equation with periodic boundary conditions on curved boundaries.

Specify a region.

In[1]:=

\[CapitalOmega] = RegionDifference[RegionUnion[Disk[], Rectangle[{0, -1}, {2, 1}]], Disk[{2, 0}]];

Solve the partial differential equation with periodic boundary conditions where the solution from the left-hand side is mapped to the right-hand side of the region.

In[2]:=

ufun = NDSolveValue[{-\!\( \*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[x, y]\)\) == 1, PeriodicBoundaryCondition[u[x, y], (x - 2)^2 + y^2 == 1, Function[x, x - {2, 0}]], DirichletCondition[ u[x, y] == 0, (0 <= x < 2 - 10^-6 && (y <= -1 || y >= 1))]}, u, {x, y} \[Element] \[CapitalOmega]];

Visualize the solution.

In[3]:=

ContourPlot[ufun[x, y], {x, y} \[Element] \[CapitalOmega], ColorFunction -> "TemperatureMap", AspectRatio -> Automatic]

Out[3]=