Konstantin,
What is the correct way to set smooth coefficients in space for high p-order
of elements simulation in deal.ii?
E.g. in Step-6 nonconstant coefficient is a step function alpha=20 for R<0.5
and alpha=1 otherwise (0<R<1). How to set it to be a smooth function, e.g.
some polynomial function of order 'n' like alpha=20*(1-R^n)+1? So it should be
enough to use few high-order elements for the whole model to converge to the
smooth solution.
I`ve checked a number of tutorials
(e.g.
http://www.dealii.org/developer/doxygen/deal.II/step_4.html#Righthandsideandboundaryvalues
, equation data section in step-27, and few others - all of them use a
point-wise approximation of an analytic function (so it is tuned for
h-refinements), which is practically the same as a step function for a given
mesh.
No, that's not true. When you say "pointwise", what I think you really mean is
that we only evaluate the coefficient at individual (quadrature) points. But
that's good enough, because when you do quadrature -- say for an integrand on
a cell K,
A_{ij}^K = \sum_q a(x_q) \nabla phi_i(x_q) \nabla phi_j(x_q) JxW(q)
then that is equivalent to computing the integral of a polynomial that is
equal to
a(x_q) \nabla phi_i(x_q) \nabla phi_j(x_q)
i.e., that *interpolates* these point values. This polynomial is of course
smooth. In particular, this corresponds to a *smooth* interpolation of the
coefficient a(x_q), even though you only evaluate it at quadrature points.
What kind of polynomial you use to interpolate the integrand depends on how
many quadrature points you use. If you need a very accurate interpolation of
the coefficient, just use a higher order Gauss formula.
So in some sense, you've got it exactly the wrong way around: If, in step-6,
we use a discontinuous coefficient, then the finite element method using
quadrature really sees only a polynomial approximation of that discontinuous
coefficient on every mesh. (Though that polynomial approximation of course
tends to a discontinuous function under mesh refinement.)
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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