why is nodes not uniformly spaced (apart from the end points)?

[dlg0]

@MollyRaver

The nodes variable, say for running the following (with a stop at line 101 of asgard.m) ...

asgard(projecti_diff1,'lev',3,'deg',3,'timestep_method','FE', 'dt',0.01,'num_steps',16)

diff(nodes{1})
ans =

   2.8175e-02
   9.6825e-02
   9.6825e-02
   5.6351e-02
   9.6825e-02
   9.6825e-02
   5.6351e-02
   9.6825e-02
   9.6825e-02
   5.6351e-02
   9.6825e-02
   9.6825e-02
   5.6351e-02
   9.6825e-02
   9.6825e-02
   5.6351e-02
   9.6825e-02
   9.6825e-02
   5.6351e-02
   9.6825e-02
   9.6825e-02
   5.6351e-02
   9.6825e-02
   9.6825e-02
   2.8175e-02

you will see non-uniform grid spacing (beyond just the end points). So this has be somewhat confused. Any thoughts on this?

[MollyRaver]

The "nodes" is generated from file "matrix_plot_D.m". On line 26, we are using Gaussian quadrature (non-uniform points) and on the first and last cells we include left and right boundary points. If we want to use uniform points we can do:

[quad_x_interior_element,quad_w] = linspace(-1,1,Num_Pts_Plot); or [quad_x_interior_element,quad_w] = linspace(-1+delta,1-delta,Num_Pts_Plot);

The first one will generate the same points but with different function values for DG and will show wiggling. The second one will move the cell boundary a little bit (depending on the value of delta), and thus deal with the wiggling plot.

[efdazedo]

If we have the function values at quadrature points, perhaps we can use interpolation to evaluate the function on another set of uniform points. The whole process may be viewed as a matrix-vector multiply. Maybe something like f_uniform = T f_gaussian, if T is square and well conditioned, perhaps inv(T) f_uniform = f_gaussian. Would this help? Just a thought.

[dlg0]

@MollyRaver There seems to be a challenge here that if we want a uniform sampling AND to include the boundary points, then we MUST sample at the interior element boundaries - which means dual values at the interior element boundaries?

Is there a "correct" way to average together the dual values at the element boundaries?

[dlg0]

I have pushed an update to reference which provided for an 'uniform_output',true option which gives the realspace output on a uniform grid. However, it "solves" the dual value at interior element boundary problem by simply picking the left element. For values of deg >=3 this has no practical issues, but below that things look strange as expected.

[MollyRaver]

Could you please comment on the code and let me reproduce the plotting? I tried with advection equation, and deg = 1 or 2 look ok to me. @dlg0

[dlg0]

OK, this is all good. Closing.