In fact, I should solve my problem on a sphere (circle). According to your
noteworthy comments, the only thing that I have to do is change the geometry
and use higher order mapping for accuracy. Another thing is that consider a
half of a circle in r-z coordinate system and the problem is axisymmetric.
what should be my boundary condition for rotation axis (z-coordinate)?
That depends on the equation, but if you were, for example, solving the
Laplace equation in cylindrical coordinates, then there would be no special
boundary condition along the z-axis (at r=0). That's because you will find
that when you do the integration by parts, you get boundary terms of the form
\int_{\partial \Omega} n.nabla u(r,z) varphi(r,z) 2*pi*r dr dz
But at the symmetry axis, r=0, so the integrand disappears at the boundary.
In other words, for most equations, there is nothing special you need to do at
the symmetry axis.
Best
W.
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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