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