This is difficult and depends on the kind of grid. For a totally arbitrary irregular grid Baumgardners method is in principle possible. However, it is computationally for higher order so expensive that Baumgardner did use it for second order only. If you can exploit some regularity of the grid, such as some of the points lining up, quite efficient methods are possible. See mon sea rev 136 no 7 2683 for this and the citation of Baumgardners work

Many thanks,

we are seeking something for arbitrary irregular grids, since we are using a priori error estimates and in future even adaptive grids, there is no chance for exploring grid regularities, however, many thanks for the hint! 

Cheers

Aron

Dear Aron,

could you please be a little more specific, by telling us where (at which points of the mesh) the data you need to derive will be given?

You may use interpolating polynomial functions of high order in R^N and then derive them to get your high-order derivatives.

This is something which is used for example in finite volume computations with high order reconstructions.

Best regards,

Pascal Omnes

Pascal

For arbitrary grids, even for 1st order elements, you can compute the L^2 projection of the derivative over the mesh, integrate by parts and solve the finite element problem. Then go on until you get to the derivative you want.

For example, to compute g = d_x G you can do:

(v,g)=(v,d_x G)

and then solve:

(v,g) = (d_x v , G) + boundary term

My inclination, at least for a start, would be to do a least-squares reconstruction.  With a sufficiently large stencil, this can give you as many derivatives as you need.  For instance, to get third derivatives in 2D to second order accuracy, you'd need a reconstruction that goes as high as fourth derivatives.  This is a total of 15 terms; I'd use about 50% more data than this so the L2 reconstruction can filter out noise.  (And yes, this requires smooth data, but that's always true for derivatives...)

See the paper below, or any of Tim Barth's stuff on reconstruction, for mathematical details of the process.

Article A High-Order-Accurate Unstructured Mesh Finite-Volume Scheme...