I would like to extend the current fe_values function computing the
Laplacian to the case where a mappingQ is given.
If I am not mistaken, this is not possible in the current version of
dealii.
Doesn't it compute second derivatives by finite differences currently,
independent of the mapping?
Of coarse the issue is to compute the derivatives of the coefficients
appearing because of the covariant derivative.
Does someone has a (clever) idea on how to proceed?
Yes. I believe Guido implemented the finite difference scheme because we
didn't see how one would compute the derivative of the inverse Jacobian
in the expression
J^{-1} \hat \nabla [ J^{-1} \hat \nabla \phi_i]
efficiently. But that's actually not so difficult because
nabla J^{-1} = -J^{-1} [nabla J] J^{-1}
with appropriate contraction over indices. In other words, one can
compute the second derivatives at a quadrature point by computing the
following:
- gradients and Hessians of the shape functions on the reference cell
- Jacobians of the mapping
- the inverses of the Jacobians of the mapping
- the gradients of the Jacobians of the mapping
I believe that in one form or another, all of this is already available.
It just needs to be plugged together to give the correct set of second
derivatives on the real cell.
Best
W.
------------------------------------------------------------------------
Wolfgang Bangerth email: [email protected]
www: http://www.math.tamu.edu/~bangerth/
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