Hi Wolfgang,
I can get a table if it would be useful.
I see what you mean in terms of convergence. I guess I was looking for the
accuracy pointwise on a boundary where the Dirichlet condition for the
pressure is imposed weakly. In my case, the value of the output on the
boundary was important, which is why I was looking at the value of the
pressure on the boundary.
I will try using Q2-Q2 elements, though I realise that this doesn't change
the application of the BC itself, so the issues may be the same, but
definitely worth a try.
For a neumann boundary condition though (my problem has that on another
boundary), am I correct in thinking that project_boundary_values doesn't
allow for component masking? the condition is only on u = f(grad p) and not
on p itself so I would need something similar to component masking the
velocities.
Many thanks
On Monday, April 15, 2019 at 3:23:28 PM UTC+1, Wolfgang Bangerth wrote:
>
>
> Jane,
>
> > I continued to find out why I wasn't getting the correct applied
> Dirichlet
> > values on the boundary for a code very similar to step-20, where the
> Dirichlet
> > condition is applied weakly using
> >
> > for (unsigned int face_no=0;
> > face_no<GeometryInfo<dim>::faces_per_cell;
> > ++face_no)
> > if (cell->at_boundary(face_no))
> > {
> > fe_face_values.reinit
> > <
> https://www.dealii.org/8.4.1/doxygen/deal.II/classFEFaceValues.html#af6e079ca7429d54433343d50bd334c3c>
>
>
> > (cell, face_no);
> > pressure_boundary_values
> > .value_list (fe_face_values.get_quadrature_points
> > <
> https://www.dealii.org/8.4.1/doxygen/deal.II/classFEValuesBase.html#a5f8732ebe2d3c6746f6de26a79cb1e45>(),
>
>
> > boundary_values);
> > for (unsigned int q=0; q<n_face_q_points; ++q)
> > for (unsigned int i=0; i<dofs_per_cell; ++i)
> > local_rhs(i) += -(fe_face_values[velocities].value (i, q) *
> > fe_face_values.normal_vector
> > <
> https://www.dealii.org/8.4.1/doxygen/deal.II/classFEValuesBase.html#a130eea0fa89263d93b20521addc830c7>(q)
>
>
> > *
> > boundary_values[q] *
> > fe_face_values.JxW
> > <
> https://www.dealii.org/8.4.1/doxygen/deal.II/classFEValuesBase.html#ad097580a2f71878695096cc73b271b9d>(q));
>
>
> > }
> >
> >
> > I then looked at step-20 - I used the exact code but solved directly
> instead,
> > giving me the same results as in the tutorial (for the errors etc).
> >
> > Even in step-20, the boundary values aren't correct for most of the
> tests, or
> > rather, have a lot of error in itself.
> > For example, at the point (1,1) which is on the boundary, p = -1.1 with
> the
> > given test problem.
> >
> > These are the values I obtained having run the code:
> > at lowest degree (0), refinement level (RL) 3, p=-0.941992, which is
> rather
> > far off the -1.1 value for the Dirichlet condition applied. even at RL6,
> p=1.07976
> > I tried the next degree up (1), at RL3, p= -1.09984. eventually at RL6
> for
> > this degree, we get -1.1.
> > Similarly, at degree 2, at RL3. p is still not accurate at p = -1.10004
>
> Individual values are usually not particularly meaningful. What matters is
> the
> convergence towards the correct value. Can you produce a table for each
> order
> in which you show the value as a function of refinement level?
>
> I will add that it's not clear to me that the finite element method
> applied to
> this problem guarantees that the error *at a single point* converges to
> zero.
> We know that the error converges to zero when measured in integral
> quantities
> such as the L2 norm, but that mean not imply pointwise convergence.
>
>
> > How else can we imposed the Dirichlet condition (without having to use
> very
> > high refinement levels and degrees) on the boundary for this problem?
>
> I don't know of other ways. The question I would ask first is whether what
> you
> have *converges*. That's all you can really hope for. The second question
> is
> whether the error is *small*. If the method converges, then the error
> *will*
> be small if only the mesh is fine enough; but "fine enough" may be smaller
> than you can computationally afford, and in that case one can start to ask
> about alternatives (such as using the Q2-Q1 element used in step-43
> instead of
> the RT element you have here), but the first step is to unambiguously
> establish convergence.
>
> Best
> WB
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
>
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