> - which kind of estimator we should use in this cases?
For mesh refinement purposes, the second derivative is probably a useful
indicator. For error estimation you would probably have to go down to less
derivatives, such as
||e||_{L^2} \le C h ||u||_{H^1}
> - it seems that refinment around singularity is a feasible, becasue we have
> concentration of information around them, e.g. when we have a point
> singularity be refinement we explore details of local solution structure,
> so we expct decrease of l2-norm, though estimator does not decrease, also
> at least most of estimators refine grid in the vicinity of singularities,
> any comment on this issue?
Yes, you'd expect the error to decrease but the estimator based on second
derivatives to not decrease. Certainly refining in the vicinity of the
singularity is a useful thing.
> - why estimator does not decrease?
Because it approximates the norm of the second derivative which does not exist
for functions with point singularities in H^1
> because at least (theoritically) we
> compute a measure of estimator inside grid and so lower dimansional
> features can not have contribution in integral, am i miss something?
The Kelly estimator computes terms on the interfaces of cells. So if these
terms are not square integrable, then the Kelly estimator should yield
something that does not converge to zero.
> also i have another kind question: it is well known that (at least proved
> for some specific problems) interpolation error make a bound for
> discritization error, so it can be considered as an error estimator (though
> is not very common, else for anisotorpic refinement), what is drawback of
> using such estimations?
You mean
||u-u_h||_{H^1} \le Cp || u - I_hu || \le Cp Ci h || u ||_{H^2}
where Ci is an interpolation constant and Cp is an approximation constant? How
do you intend to evaluate the right hand side here? If you approximate the
second derivatives of u on the right hand side using jumps of normal
components of the gradient across cell faces then you get exactly the Kelly
estimator.
W.
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Wolfgang Bangerth email: [EMAIL PROTECTED]
www: http://www.math.tamu.edu/~bangerth/
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