Define and refine mesh around a hole

Question

I am solving Laplace equation with particular geometry and boundary conditions:

L = 10; a = 1; k = 1/2; mesh = 0.06;

BoxL = x == L || x == -L || y == L || y == -L;

reg = ImplicitRegion[(-L <= x <= L && -L <= y <= L) && (x^2 + y^2 >=  
  a^2), {x, y}];
rp = RegionPlot[reg, AspectRatio -> Automatic, ImageSize -> Tiny]

Boundary condition for θ

bcθ = {DirichletCondition[u[x, y] == ArcTan[x, y] k, x^2 + y^2 == a^2],
   DirichletCondition[u[x, y] == 0, BoxL]};

Solving equation for θ

θ = NDSolveValue[{Laplacian[u[x, y], {x, y}] == 0, bcθ}, 
 u, {x, y} ∈ reg, Method -> {"FiniteElement", "MeshOptions" -> {MaxCellMeasure -> mesh}, 
"InterpolationOrder" -> {u -> 2}}];

And then plot:

Magnifying and showing the mesh:

There appears to be some small imperfections. Of course I have tuned the parameter mesh to minimize them, i.e., the mesh "scale" is the best I could find within this "shape".

Here comes my question: how do I define a mesh which is finer around the central circular hole (which is a boundary)? Is there a simple way without going down to draw it?

Thanks!

Answer 1

You can use "MaxBoundaryCellMeasure" for that:

L = 10; a = 1; k = 1/2; mesh = 0.06;
BoxL = x == L || x == -L || y == L || y == -L;
reg = ImplicitRegion[(-L <= x <= L && -L <= y <= L) && (x^2 + y^2 >= a^2), {x, y}];
bcθ = {DirichletCondition[u[x, y] == ArcTan[x, y] k, x^2 + y^2 == a^2], 
   DirichletCondition[u[x, y] == 0, BoxL]};

θ = NDSolveValue[{Laplacian[u[x, y], {x, y}] == 0, bcθ}, 
   u, {x, y} ∈ reg, 
   Method -> {"FiniteElement", "MeshOptions" -> {"MaxBoundaryCellMeasure" -> mesh}}];

Show[ContourPlot[θ[x, y], {x, y} ∈ θ["ElementMesh"], 
  PlotRange -> All], θ["ElementMesh"]["Wireframe"], PlotRange -> {{0, 2}, {-1, 1}}]