Isoparametric testing

[Lukas Bystricky]

I'm trying to test the convergence rates of isoparametric elements for the Poisson equation. To do this I have a circular domain and an exact solution. My understanding is that with linear mappings I should not get optimal convergence in the L2 and H1 norms for elements higher than degree 1. However when I test this, I recover the optimal accuracy with quadratic, cubic and quartic elements.

Here is my code to construct the mesh (using version 8.2.1)

Point<2> center (0,0);
double radius = 1;
GridGenerator::hyper_ball(

mesh, center, radius);
static const HyperBallBoundary<dim> boundary_description(center);
mesh.set_boundary(0, boundary_description);

I think that what's happening is that my computational domain is a polygon (using linear mappings) and when I apply the exact solution as a boundary condition deal.ii knows the exact value at all points along my polygonal boundary. In reality, it should only know the values on the circle and do some interpolation to figure out the values along the polygon. Here is my code to apply the boundary conditions:

Am I correct in this assumption? If so, is there a good way to test the convergence rates and demonstrate sub-optimal convergence for linear mappings?

[Timo Heister]

> Am I correct in this assumption? If so, is there a good way to test the
> convergence rates and demonstrate sub-optimal convergence for linear
> mappings?

Yes, you are just checking convergence of a changing polygonal domain.
I did the same mistake not too long ago. :-)

One way to do this is to design a solution that is 0 on the circle and
apply homogeneous boundary conditions instead of using the exact
solution for interpolating the boundary values.

--
Timo Heister
http://www.math.clemson.edu/~heister/

[Lukas Bystricky]

I guess I'm in good company then. ;-)

That fixed my problem. Thanks!

[Wolfgang Bangerth]

On 03/05/2017 05:19 PM, Lukas Bystricky wrote:
> I'm trying to test the convergence rates of isoparametric elements for the
> Poisson equation. To do this I have a circular domain and an exact solution.
> My understanding is that with linear mappings I should not get optimal
> convergence in the L2 and H1 norms for elements higher than degree 1. However
> when I test this, I recover the optimal accuracy with quadratic, cubic and
> quartic elements.

I haven't tried very hard to understand your question because Timo has already
answered, but I'd like to point out that odd order approximations of the
geometry are as good as the ones with one degree lower (even) approximations
of the geometry for your particular domain: a circle. That is, because for a
circle, the circle's arc over a face is symmetric, and so has only even
components if you try to approximate it by a polynomial arc.

A similar effect is mentioned in one of the tutorial programs that deal with
mappings (9 and 10? or 10 and 11?).

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/