[Geopdes-users] k-refinement oscilations in convergence rate

[alexis p. <ale...@gm...>]

Dear geopdes-users,

I tried to test some convergenece rates by applying several refinement
strategies such k-refinement and h refinement.Although h refinement has
relatively a smooth convergence , krefinement presents a lot of oscillations
.After creating a krefinement m file as such:

function geo = krefinement( geo,degr,knts)

nurbs=geo.nurbs;
degelev=max(degr-(nurbs.order-1),0);
nurbs=nrbdegelev(nurbs,degelev);
[rknots,zeta,nknots]=kntrefine(nurbs.knots,knts,nurbs.order-1,nurbs.order-2);
nurbs=nrbkntins(nurbs,nknots);
geo=geo_load(nurbs);


end



I then apply homo_kerror.m to the 'geo_ring.txt' domain

function errors = homo_kerror(geometry,i )

errors=[];
for j=1:1:i
    [dofs(1) errors(1)] = homo_poisson( geometry );
    refined_geo(j)=krefinement(geometry,[j j],[1 1]);
     [dofs(j+1) errors(j+1)]=homo_poisson(refined_geo(j));


   [dofs3(1) errors3(1)] = homo_poisson( geometry );
    refined_geo2(j)=hrefinement(geometry,[j j]);
   [dofs2(j+1) errors2(j+1)]=homo_poisson(refined_geo2(j));




end
         loglog(dofs,errors,'-ko',dofs2,errors2,'-go')
         legend('krefinement','hrefinement')
         xlabel('dofs');
 ylabel('error');
         grid on
end

where homo_poisson.m is the problem presented in the geopdes report (article
4.1)

function [dofs ,err] = homo_poisson( geometry )
nurbs=geometry.nurbs;
geometry=geo_load(nurbs);
knots = geometry.nurbs.knots;

[qn, qw] = msh_set_quad_nodes (knots, msh_gauss_nodes
(geometry.nurbs.order));
msh = msh_2d (knots, qn, qw, geometry);

space  = sp_nurbs_2d (geometry.nurbs, msh);
dofs=space.ndof;
mat = op_gradu_gradv_tp (space, space, msh, @(x, y) ones (size (x)));
rhs = op_f_v_tp (space, msh, @(x, y)
(8-9*sqrt(x.^2+y.^2)).*sin(2*atan(y./x))./(x.^2+y.^2));

drchlt_dofs = [];
for iside = 1:4
  drchlt_dofs = union (drchlt_dofs, space.boundary(iside).dofs);
end
int_dofs = setdiff (1:space.ndof, drchlt_dofs);

u = zeros (space.ndof, 1);
u(int_dofs) = mat(int_dofs, int_dofs) \ rhs(int_dofs);

sp_to_vtk (u, space, geometry, [20 20], 'laplace_solution.vts', 'u')
err = sp_l2_error (space, msh, u,
@(x,y)(x.^2+y.^2-3*sqrt(x.^2+y.^2)+2).*sin(2.*atan(y./x)))




end

I tried several domains and problems and I get similar results. In
krefinement the order elevates by 1 and then a new knot is inserted(in both
directions) at each step. any suggestions why this occurs?

Thank you,
Alexis Papagiannopoulos