DirichletBC expressions evaluated at the interior

(From http://fenicsproject.org/qa/1033/dirichletbc-expressions-evaluated-away-from-the-boundary.)

When using an Expression for Dirichlet boundary conditions, this expression is not only evaluated at boundary points but also on the interior on the domain. This should be avoided since Expression might be very expensive to compute, or might not even be properly defined anywhere else except on the boundary.

MWE:

from dolfin import *

from matplotlib import pyplot as pp

class MyExpensiveBoundaryExpression(Expression):
    def eval(self, values, x):
        values[0] = 0.0
        pp.plot(x[0], x[1], 'xk')
        return

my_expr = MyExpensiveBoundaryExpression()

mesh = UnitSquareMesh(10, 10)
V = FunctionSpace(mesh, 'CG', 2)

u = TrialFunction(V)
v = TestFunction(V)

a = dot(grad(u), grad(v)) * dx
L = v * dx

bcs = DirichletBC(V, my_expr, 'on_boundary')

sol = Function(V)
solve(a == L, sol, bcs=bcs)

pp.show()

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