Name

Another Laplacian with exact solution — This is the scilab/demos/demo_laplacian.sce example.

Description

 
gf_workspace('clear all');
m = gf_mesh('cartesian',[0:.1:1],[0:.1:1]);

// create a mesh_fem of for a field of dimension 1 (i.e. a scalar field)
mf = gf_mesh_fem(m,1);
// assign the Q2 fem to all convexes of the mesh_fem,
gf_mesh_fem_set(mf,'fem',gf_fem('FEM_QK(2,2)'));

// an exact integration will be used
mim = gf_mesh_im(m, gf_integ('IM_GAUSS_PARALLELEPIPED(2,10)'));
// detect the border of the mesh
border = gf_mesh_get(m,'outer faces');
// mark it as boundary #1
gf_mesh_set(m, 'boundary', 1, border);
gf_plot_mesh(m, 'regions', [1]); // the boundary edges appears in red

// interpolate the exact solution
Uexact = gf_mesh_fem_get_eval(mf, list('y.*(y-1).*x.*(x-1)+x.^5'));
// its second derivative
F      = gf_mesh_fem_get_eval(mf, list('-(2*(x.^2+y.^2)-2*x-2*y+20*x.^3)'));

md=gf_model('real');
gf_model_set(md, 'add fem variable', 'u', mf);
gf_model_set(md, 'add Laplacian brick', mim, 'u');
gf_model_set(md, 'add initialized fem data', 'VolumicData', mf, F);
gf_model_set(md, 'add source term brick', mim, 'u', 'VolumicData');
gf_model_set(md, 'add initialized fem data', 'DirichletData', mf, Uexact);
gf_model_set(md, 'add Dirichlet condition with multipliers', mim, 'u', mf, 1, 'DirichletData');

gf_model_get(md, 'solve');
U = gf_model_get(md, 'variable', 'u');

printf('H1 norm of error: %g\n', gf_compute(mf,U-Uexact,'H1 norm',mim));

subplot(2,1,1); 
gf_plot(mf,U,'mesh','on','contour',.01:.01:.1); 
colorbar(min(U),max(U)); 
title('computed solution');

subplot(2,1,2); 
gf_plot(mf,U-Uexact,'mesh','on'); 
colorbar(min(U-Uexact),max(U-Uexact));
title('difference with exact solution');
 

See Also

gf_workspace, gf_mesh, gf_fem, gf_plot

Authors

Y. Collette