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From: Narek Saribekyan on 12 May 2010 02:27
Hello,
Suppose u is in H^l_0 on bounded region (anisotropic Sobolev space,
l=(l1,...ln) - multi-index).
Note that:
All partial (but not mixed) derivatives of u op to orders l_i vanish
on the boundary.

I'm trying to construct a finite element basis for H^l_0. Using
anisotropic Strang-Fix conditions for choosing basis elements, it
turns out that multivariate B-splines are appropriate for that purpose
(product of univariate B-splines of order l_i for each direction). The
problem arises near the boundary, because finite elements near the
boundary don't vanish on the boundary (although, their supports have
non-empty intersection with the region). I was thinking about getting
rid of these elements by taking only those, which have supports
completely inside the region. To get an estimate, I need an estimate
of norm near the boundary. But u doesn't have mixed-partial
derivatives at all, therefore, no Taylor expansion is possible. But
maybe there's an estimate in case of Sobolev norm?

I don't know if I made it clear to you.
Thanks a lot,
Narek


From: Han de Bruijn on 12 May 2010 03:12
On May 12, 8:27 am, Narek Saribekyan <narek.saribek...(a)gmail.com>
wrote:
> Hello,
> Suppose u is in H^l_0 on bounded region (anisotropic Sobolev space,
> l=(l1,...ln) - multi-index).
> Note that:
> All partial (but not mixed) derivatives of u op to orders l_i vanish
> on the boundary.
>
> I'm trying to construct a finite element basis for H^l_0. Using
> anisotropic Strang-Fix conditions for choosing basis elements, it
> turns out that multivariate B-splines are appropriate for that purpose
> (product of univariate B-splines of order l_i for each direction). The
> problem arises near the boundary, because finite elements near the
> boundary don't vanish on the boundary (although, their supports have
> non-empty intersection with the region). I was thinking about getting
> rid of these elements by taking only those, which have supports
> completely inside the region. To get an estimate, I need an estimate
> of norm near the boundary. But u doesn't have mixed-partial
> derivatives at all, therefore, no Taylor expansion is possible. But
> maybe there's an estimate in case of Sobolev norm?
>
> I don't know if I made it clear to you.
> Thanks a lot,
> Narek

Maybe you're on the wrong track? Or rather: completely at lost?

Han de Bruijn
From: Narek Saribekyan on 12 May 2010 07:22
On May 12, 12:12 pm, Han de Bruijn <umum...(a)gmail.com> wrote:
> On May 12, 8:27 am, Narek Saribekyan <narek.saribek...(a)gmail.com>
> wrote:
>
>
>
>
>
> > Hello,
> > Suppose u is in H^l_0 on bounded region (anisotropic Sobolev space,
> > l=(l1,...ln) - multi-index).
> > Note that:
> > All partial (but not mixed) derivatives of u op to orders l_i vanish
> > on the boundary.
>
> > I'm trying to construct a finite element basis for H^l_0. Using
> > anisotropic Strang-Fix conditions for choosing basis elements, it
> > turns out that multivariate B-splines are appropriate for that purpose
> > (product of univariate B-splines of order l_i for each direction). The
> > problem arises near the boundary, because finite elements near the
> > boundary don't vanish on the boundary (although, their supports have
> > non-empty intersection with the region). I was thinking about getting
> > rid of these elements by taking only those, which have supports
> > completely inside the region. To get an estimate, I need an estimate
> > of norm near the boundary. But u doesn't have mixed-partial
> > derivatives at all, therefore, no Taylor expansion is possible. But
> > maybe there's an estimate in case of Sobolev norm?
>
> > I don't know if I made it clear to you.
> > Thanks a lot,
> > Narek
>
> Maybe you're on the wrong track? Or rather: completely at lost?
>
> Han de Bruijn

I think I'm not.
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