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Anders Logg wrote:
On Wed, Sep 16, 2009 at 04:34:29PM +0200, Garth N. Wells wrote:Anders Logg wrote:@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.I don't agree at all. The function 1-x-y can be evaluated anywhere, but the basis functions are defined only on the cell. If the basis functions were defined everywhere, the method would loose all sparsity.The sparsity is not a result of evaluate_basis returning 0 outside the cells, it's a result of the assembly process and the local-to-global mapping. Either way, that's not how evaluate_basis is implemented today. It works perfectly fine to evaluate the natural extensions of the basis functions, except along one line where the mapping happens to be singular. This is not in anyway connected to how the basis functions should be defined, it's just a consequence of the particular way Rob has implemented the basis functions in FIAT (via a mapping from Jacobi polynonials on a square). For example, if you take the first basis function (1 - x - y) on the reference triangle and evaluate it at (2, 2), you get -2 as expected although that point is outside the triangle. But if you try to evaluate it at (1, 1), you get 0 because of a bug/feature in evaluate_basis. If you do (1, 1 + eps), then you get -1 - eps.
The sparsity is a result of the basis being non-zero only locally, so I would call getting zero outside of the cell from ufc::evaluate_basis(..) a feature.
Garth
-- AndersGarth@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.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? -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.9 (GNU/Linux) iEYEARECAAYFAkqw71UACgkQTuwUCDsYZdEIagCeK7i8b18hMN/aD1tQioosHu6L GVsAn28SJ86qU+zUm2AUuYLpvTgTD7ds =qOFA -----END PGP SIGNATURE----- _______________________________________________ FFC-dev mailing list FFC-dev@xxxxxxxxxx http://www.fenics.org/mailman/listinfo/ffc-dev _______________________________________________ FFC-dev mailing list FFC-dev@xxxxxxxxxx http://www.fenics.org/mailman/listinfo/ffc-dev------------------------------------------------------------------------ _______________________________________________ FFC-dev mailing list FFC-dev@xxxxxxxxxx http://www.fenics.org/mailman/listinfo/ffc-dev_______________________________________________ FFC-dev mailing list FFC-dev@xxxxxxxxxx http://www.fenics.org/mailman/listinfo/ffc-dev
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