Hello Prof. Bangerth,
Thank you very much for the explanation. It really helps to understand FEM
and the dealii library.
I am glad to find this group to discuss things on FEM and dealii.
Best,
Yiyang
On Thursday, September 21, 2017 at 9:45:48 AM UTC-5, Wolfgang Bangerth
wrote:
>
>
> > Suppose we need to solve some PDE about some function u(r). Following
> the weak
> > form of an equation, we obtain and solve a linear system (A*x=b). What
> we
> > really solve are the coefficients U_j (this corresponds to the vector x
> in the
> > linear system). Thus I am assuming that the solution vector (normally
> denoted
> > as "system_solution" in the Tutorials) is really the vector U_j.
>
> Correct.
>
>
> > The function u(r) = \sum_j U_j Q(r)_j, where Q(r)_j are the base
> functions.
> > For a certain point in the real space, denoted as r0, the value of the
> > function u should be u(r_0) = \sum_j U_j Q(r_0)_j.
> > Thus, when I loop over the elements of the solution vector, that is, a
> loop like
> > for(unsigned int i=0; i<dof_handler.n_dofs(), ++i){
> > system_solution(i) = /*some code*/; //each i corresponds to a
> support point
> > }
> > That modifies the coefficients U_j.
> >
> > If I want to limit the value of u(r) in the range [0,1], for example.
> Let's
> > say, in a "discrete" version, my aim is to limit u(r) at every support
> point
> > in the range [0,1]. However, if I do the above loop, and limit
> > system_solution(i) in the range [0,1], how would that help to limit
> u(r),
> > since u(r) is a sum (u(r) = \sum_j U_j Q(r)_j). It looks like there are
> some
> > properties concerning the base functions and support points? Is it some
> > general properties for base functions?
>
> Yes, this is an excellent observation.
>
> There is a simple observation that helps you if you use Q_1 elements
> (piecewise (bi-/tri-)linear), namely that the maximum and minimum of u(r)
> are
> attained at the nodes (support points) of the mesh. This is easy to see if
> you
> just think of the 1d situation where u(r) is a piecewise linear function
> composed of line segments -- their maxima and minima are clearly at the
> nodes.
> The same is true in 2d/3d.
>
> But, if you are using quadratic (Q_2) elements, then this property is no
> longer true. In other words, limiting system_solution(i) does not
> guarantee
> that u(r) is also limited. There is no simple solution to this problem
> when
> using higher order Lagrange elements. I believe that I heard people say
> that
> this is easier to achieve if you use a Bernstein basis instead of a
> Lagrange
> basis, but I cannot say that I know these bases well enough to explain.
>
> Best
> W.
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
>
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