Error in definition of PeriodicBoundaryCondition?

Question

The documentation for PeriodicBoundaryCondition (https://reference.wolfram.com/language/ref/PeriodicBoundaryCondition.html) has: Where it says $u ( x_{target} ) = a + b\ u ( f ( x_{target} ) )$, I think it should instead say $u( f ( x_{target} ) ) = a + b\ u ( x_{target} )$. I could be wrong but I believe this is demonstrated by this example: https://wolfram.com/xid/0bswu24h9fy656tmxe-jnf5k3. I have copied the code here, only modifying it from $a=-1/20$ to $a=0$ because this will demonstrate what I am talking about:

Ω = Rectangle[{0, 0}, {2, 1}];
pde = -\!\(
\*SubsuperscriptBox[\(∇\), \({x, y}\), \(2\)]\(u[x, y]\)\) == 
   If[1.25 <= x <= 1.75 && 0.25 <= y <= 0.5, 1., 0.];
Subscript[Γ, D] = 
  DirichletCondition[u[x, y] == 0, (y == 0 || y == 1) && 0 < x <= 2];
a = 0; b = 2;
pbc = PeriodicBoundaryCondition[a + b*u[x, y], x == 0 && 0 <= y <= 1, 
   TranslationTransform[{2, 0}]];
ufun = NDSolveValue[{pde, pbc, Subscript[Γ, D]}, 
   u, {x, y} ∈ Ω];
ContourPlot[ufun[x, y], {x, y} ∈ Ω, 
 ColorFunction -> "TemperatureMap", AspectRatio -> Automatic]

The left-hand edge, from $(0,0)$ to $(0,1)$, is the target (i.e. where the predicate in PeriodicBoundaryCondition is true) and the right-hand edge, from $(2,0)$ to $(2,1)$, is the source because $x_{source} = f (x_{target})$. Now as $b=2$ (and $a=0$) by the current definition we would expect $u(x_{target}) = 2 u(x_{source})$, meaning the values at the left-hand edge should be twice as large as those on the right-hand edge. But they're not. Instead, they are half as large. This implies that the definition should instead be $u( f ( x_{target} ) ) = a + b u ( x_{target} )$. I have explored this for many hours with many examples and keep arriving at the same conclusion.

Answer 1

I think you analysis is correct - this is a typo in the documentation and I have updated the documentation. Sorry for the the trouble and thanks for reporting this. You always have the option to report things like this to support AT wolfram.com. I may not see all issues I am responsible for if posted here.

Answer 2

As suggested in the comments and elaborated in my answer 223465, you can use a triangle mesh and symmetrize the PeriodicBoundaryCondition by making the following workflow:

Create Triangle Mesh

Here we use ToElementMesh to create a triangle mesh with refinement on the boundaries.

Needs["NDSolve`FEM`"]
Ω = Rectangle[{0, 0}, {2, 1}];
(* Create Triangle Mesh *)
mesh = ToElementMesh[Ω, 
   "MaxCellMeasure" -> {"Length" -> 0.05}, 
   "MaxBoundaryCellMeasure" -> 0.0025, 
   "MeshElementType" -> TriangleElement];

Create a Plotting Function

Here we will create a function that simulates a parametric function, generates a contour plot, and an error plot of the periodic condition of the two boundaries.

plotFn[a_, b_][pfun_] := 
 Module[{ufun, uRange, legendBar, options, cp, error, assoc},
  ufun = pfun[a, b];
  uRange = MinMax[ufun["ValuesOnGrid"]];
legendBar = 
   BarLegend[{"TemperatureMap", uRange}, 50, 
    LegendLabel -> Style["u", Opacity[0.6`]]];
options = {PlotRange -> uRange, 
    ColorFunction -> ColorData[{"TemperatureMap", uRange}], 
    ContourStyle -> Opacity[0.1`], ColorFunctionScaling -> False, 
    Contours -> 30, PlotPoints -> All, FrameLabel -> {"x", "y"}, 
    PlotLabel -> 
     Style[StringTemplate["u(x,y) Field for a=`` and b=`` "][a, b], 
      18], AspectRatio -> Automatic, ImageSize -> 500};
  cp = Legended[
    ContourPlot[ufun[x, y], {x, y} \[Element] ufun["ElementMesh"], 
     Evaluate[options]], legendBar];
  cp = Rasterize@cp;
  error = 
   Plot[{a + b*ufun[0, y] - ufun[2, y]}, {y, 0, 1}, PlotPoints -> 200,
     PlotRange -> 1.*^-15 {-1, 1}];
  assoc = <|"cp" -> cp, "error" -> error|>
  ]

Set up ParametricNDSolveValue

It would be nice to look at the effect of the $a$ and $b$ parameters. So, let's use ParametricNDSolveValue to generate a parametric function so we can quickly test the parameters.

pde = -Laplacian[u[x, y], {x, y}] == 
      If[1.25 <= x <= 1.75 && 0.25 <= y <= 0.5, 1., 0.];
ΓD = 
    DirichletCondition[u[x, y] == 0, (y == 0 || y == 1) && 0 < x < 2];
(* Symmetrized PBCs *)
pbcf = PeriodicBoundaryCondition[a + b*u[x, y], x == 0 && 0 <= y <= 1, 
      TranslationTransform[{2, 0}]];
pbcr = PeriodicBoundaryCondition[-a /b + 1/b*u[x, y], 
   x == 2 && 0 <= y <= 1, 
      TranslationTransform[{-2, 0}]];
pfun = ParametricNDSolveValue[{pde, pbcf, pbcr, ΓD}, 
     u, {x, y} ∈ mesh, {a, b}]

Test Several $a$ and $b$ values

sim01 = plotFn[0, 1][pfun]
sim02 = plotFn[0, 2][pfun]
sim03 = plotFn[1/10, 1][pfun]
sim04 = plotFn[1/10, 2][pfun]

The error between the left and right side is quite low. Introducing the offset parameter $a$ causes some ringing at the corner points. The ringing is most likely due to an inconsistency with the DirichletCondition, DC, and the PeriodicBoundaryCondition, PBC. The DC specifies zero on the top and bottom boundaries, but the PBC specifies an offset between the left and right boundary. At the corner points, there is a discontinuity between the DC and PBC.