And the nonlinear equations of elasticity are actually elliptic equations?
Yes. If the stress-strain relationship is monotonic, i.e., if the stress
increases with increasing strain. (So no bizarre materials with negative
compressibility, for example.)
-Is the *expected rate* the rate which the laplace equation (assuming that
this holds for elasticity as just discussed) yields for the gradients of the
solution, i.e. ||grad (u-u_h)||_L2 <= ...h^p ?
Yes. You'd expect h^p for the gradients, and the ZZ approach can yield
something like h^{p+1} at least in certain points. That is, you'd get
something like
|| r - nabla u || + O(h^{p+1})
where r is the reconstructed gradient. (Assuming your coefficient is 1.
Otherwise you'd have something like
|| r - kappa nabla u || + O(h^{p+1})
where r is the reconstructed stress and kappa is the stress strain tensor.
For me it's not yet fully clear how convergence rate of a projection method
(L2, SPR) is linked to ||grad (u-u_h)||_L2 <= ...h^p .
Basically the input for the projection methods are the gradients (stresses) at
the qps. Of course these values will be changed by the method( apply
projection -> get_function_values), but is this "change" totally independent
of the rate p from ||grad (u-u_h)||_L2 <= ...h^p ?
When you say "these values will be changed", can you explain what you refer
to? What is "before" and what is "after" this change?
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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