@Garth: Yes, they are defined everywhere. For example, v1 = 1 - x - y,
v2 = x and v3 = y are a nodal basis for the reference triangle, but
they are also a basis for P1 on R^2. You can evaluate 1 - x - y for
any values of x and y.
@Matt: We use the local basis on a triangle K as a basis for P2 (and
other spaces) on a patch of elements surrounding K. We make an Ansatz
in terms of the local basis functions. So it's not for extrapolation
of a computed solution.
We could also use the Bernstein polynomials on K or any other
reasonable well-conditioned basis, but it would be convenient to use
the basis that's already in there.
--
Anders
On Wed, Sep 16, 2009 at 09:05:20AM -0500, Matthew Knepley wrote:
Isn't this strange since the error bounds no longer apply outside the element
(it is no longer interpolation, but extrapolation)?
Matt
On Wed, Sep 16, 2009 at 8:59 AM, Anders Logg <logg@xxxxxxxxx> wrote:
Marie and I came across a "bug" when trying to use evaluate_basis in a
way it was probably not intended for.
In particular, we want to evaluate the basis functions for a cell in a
point that lies outside the cell. This fails when we evaluate the
basis functions in a point where the collapsing square map in FIAT is
singular.
Here's a simple script that illustrates the problem:
from dolfin import *
import numpy
mesh = UnitSquare(1, 1, "left")
V = FunctionSpace(mesh, "CG", 1)
element = V.dolfin_element()
cell = Cell(mesh, 0)
value = numpy.zeros(1)
# Value should be 1.0 (constant along x = 1)
element.evaluate_basis(1, value, numpy.array((1.0, 1.0000000000001)),
cell)
print value
# Should also be 1.0 but result is 0 (eps)
element.evaluate_basis(1, value, numpy.array((1.0, 1.0000000000000)),
cell)
print value
The problem is the following lines in evaluate_basis:
// Map coordinates to the reference square
if (std::abs(y - 1.0) < 1e-14)
x = -1.0;
else
x = 2.0 *x/(1.0 - y) - 1.0;
y = 2.0*y - 1.0;
In the above example, y = 1 so x is mapped to -1.0 on Rob's reference
square.
Is there a simple fix that would allow us to evaluate the basis
functions outside the element?
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