Managed I managed to figure it out.  Below is an example where I have a domain with 5 boundaries and I am trying to set the Physical Line like someone suggested.  Does this look right?

If yes, then I think I'm in good shape.  One problem is that the boundary layer is very narrow for my choice of parameters so it's hard to know whether the boundary value is really zero.  I don't suppose there is an easy way to find out the maximum of the boundary value in the numerical solution?

element_size = 0.05;

// Printing points from 1 to 6.

Point(1) = {-0.000000, -0.000000, 0, element_size};

Point(2) = {4.27, 0.54, 0, element_size};

Point(3) = {2.98, 4.00, 0, element_size};

Point(4) = {0.50, 3.58, 0, element_size};

Point(5) = {-1.70, 1.33, 0, element_size};

// Printing Spline points 1 to 6.

Line(1) = {1,2};

Line(2) = {2,3};

Line(3) = {3,4};

Line(4) = {4,5};

Line(5) = {5,1};

// Printing line loop from spline 1 to 2

Line Loops(1) = {1,2,3,4,5};

Plane Surface(6) = {1};

Physical Surface(7) = {6};

Physical Line(8) = {1};