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% Copyright (C) 2015-2016 Yves Renard.
%
% This file is a part of GetFEM++
%
% GetFEM++ is free software; you can redistribute it and/or modify it
% under the terms of the GNU Lesser General Public License as published
% by the Free Software Foundation; either version 3 of the License, or
% (at your option) any later version along with the GCC Runtime Library
% Exception either version 3.1 or (at your option) any later version.
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
% License and GCC Runtime Library Exception for more details.
% You should have received a copy of the GNU Lesser General Public License
% along with this program; if not, write to the Free Software Foundation,
% Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
% A Poisson problem solved with a Discontinuous Galerkin method (or
% interior penalty method). See for instance
% "Unified analysis of discontinuous Galerkin methods for elliptic
% problems", D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, SIAM J.
% Numer. Anal. vol. 39:5, pp 1749-1779, 2002.
% Options for prescribing the Dirichlet condition
dirichlet_version = 3; % 0 = simplification, 1 = with multipliers,
% 2 = penalization, 3 = Nitsche's method
theta = 1; % Nitsche's method parameter theta
gamma0 = 0.001; % Nitsche's method parameter gamma0 (gamma = gamma0*h)
r = 1e8; % Penalization parameter for the Dirichlet condition
draw = true;
quadrangles = true;
NX = 20;
K = 2; % Degree of the discontinuous finite element method
interior_penalty_factor = 300*NX; % Parameter of the interior penalty term
verify_neighbour_computation = true;
asize = size(who('automatic_var654'));
if (asize(1)) draw = false; end;
% trace on;
gf_workspace('clear all');
if (quadrangles)
m = gf_mesh('cartesian',[0:1/NX:1],[0:1/NX:1]);
else
m=gf_mesh('import','structured',sprintf('GT="GT_PK(2,1)";SIZES=[1,1];NOISED=0;NSUBDIV=[%d,%d];', NX, NX));
end
% Create a mesh_fem of for a field of dimension 1 (i.e. a scalar field)
mf = gf_mesh_fem(m,1);
% Assign the QK or PK fem to all convexes of the mesh_fem, and define an
% integration method
if (quadrangles)
gf_mesh_fem_set(mf,'fem',gf_fem(sprintf('FEM_QK_DISCONTINUOUS(2,%d)', K)));
mim = gf_mesh_im(m, gf_integ('IM_GAUSS_PARALLELEPIPED(2,6)'));
else
gf_mesh_fem_set(mf,'fem',gf_fem(sprintf('FEM_PK_DISCONTINUOUS(2,%d)', K)));
mim = gf_mesh_im(m, gf_integ('IM_TRIANGLE(6)'));
end
% Detect the border of the mesh
border = gf_mesh_get(m,'outer faces');
GAMMAD=1;
gf_mesh_set(m, 'region', GAMMAD, border);
% Inner edges for the interior penalty terms
in_faces = gf_mesh_get(m,'inner faces');
INNER_FACES=2;
gf_mesh_set(m, 'region', INNER_FACES, in_faces);
if (verify_neighbour_computation)
TEST_FACES=3;
adjf = gf_mesh_get(m, 'adjacent face', 43, 1);
if (size(adjf,2) == 0)
error('Adjacent face not found');
end
gf_mesh_set(m, 'region', TEST_FACES, [[43; 1], adjf]);
if (draw)
figure(2);
gf_plot_mesh(m, 'regions', [TEST_FACES], 'convexes', 'on');
figure(1)
end
end
% Mesh plot
if (draw)
subplot(2,2,1); gf_plot_mesh(m, 'regions', [GAMMAD]); % the boundary edges appear in red
subplot(2,2,2); gf_plot_mesh(m, 'regions', [INNER_FACES]); % inner edges appear in red
pause(1);
end
% Interpolate the exact solution
% Uexact = gf_mesh_fem_get(mf, 'eval', { '10*y.*(y-1).*x.*(x-1)+10*x.^5' });
Uexact = gf_mesh_fem_get(mf, 'eval', { 'x.*sin(2*pi*x).*sin(2*pi*y)' });
% Opposite of its laplacian
% F = gf_mesh_fem_get(mf, 'eval', { '-(20*(x.^2+y.^2)-20*x-20*y+200*x.^3)' });
F = gf_mesh_fem_get(mf, 'eval', { '4*pi*(2*pi*x.*sin(2*pi*x) - cos(2*pi*x)).*sin(2*pi*y)' });
md=gf_model('real');
gf_model_set(md, 'add fem variable', 'u', mf);
gf_model_set(md, 'add linear generic assembly brick', mim, 'Grad_u.Grad_Test_u');
% 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);
switch (dirichlet_version)
case 0,
gf_model_set(md, 'add Dirichlet condition with simplification', 'u', GAMMAD, 'DirichletData');
case 1,
gf_model_set(md, 'add Dirichlet condition with multipliers', mim, 'u', mf, GAMMAD, 'DirichletData');
case 2,
gf_model_set(md, 'add Dirichlet condition with penalization', mim, 'u', r, GAMMAD, 'DirichletData');
case 3,
gf_model_set(md, 'add initialized data', 'gamma0', [gamma0]);
expr = gf_model_get(md, 'Neumann term', 'u', GAMMAD);
gf_model_set(md, 'add Dirichlet condition with Nitsche method', mim, 'u', expr, 'gamma0', GAMMAD, theta, 'DirichletData');
end
% Interior penalty terms
gf_model_set(md, 'add initialized data', 'alpha', [interior_penalty_factor]);
jump = '((u-Interpolate(u,neighbour_elt))*Normal)';
test_jump = '((Test_u-Interpolate(Test_u,neighbour_elt))*Normal)';
grad_mean = '((Grad_u+Interpolate(Grad_u,neighbour_elt))*0.5)';
grad_test_mean = '((Grad_Test_u+Interpolate(Grad_Test_u,neighbour_elt))*0.5)';
% gf_model_set(md, 'add linear generic assembly brick', mim, sprintf('-((%s).(%s))', grad_mean, test_jump), INNER_FACES);
% gf_model_set(md, 'add linear generic assembly brick', mim, sprintf('-((%s).(%s))', jump, grad_test_mean), INNER_FACES);
% gf_model_set(md, 'add linear generic assembly brick', mim, sprintf('alpha*((%s).(%s))', jump, test_jump), INNER_FACES);
gf_model_set(md, 'add linear generic assembly brick', mim, sprintf('-((%s).(%s))-((%s).(%s))+alpha*((%s).(%s))', grad_mean, test_jump, jump, grad_test_mean, jump, test_jump), INNER_FACES);
gf_model_get(md, 'solve', 'noisy');
U = gf_model_get(md, 'variable', 'u');
if (draw)
subplot(2,2,3); gf_plot(mf,U,'mesh','off');
colorbar; title('computed solution');
subplot(2,2,4); gf_plot(mf,Uexact-U,'mesh','on');
colorbar;title('difference with exact solution');
end
err = gf_compute(mf, Uexact-U, 'H1 norm', mim);
disp(sprintf('H1 norm of error: %g', err));
if (verify_neighbour_computation)
A=gf_asm('generic', mim, 1, 'u*Test_u*(Normal.Normal)', TEST_FACES, md);
B=gf_asm('generic', mim, 1, '-Interpolate(u,neighbour_elt)*Interpolate(Test_u,neighbour_elt)*(Interpolate(Normal,neighbour_elt).Normal)', TEST_FACES, md);
err_v = norm(A-B);
A=gf_asm('generic', mim, 1, '(Grad_u.Normal)*(Grad_Test_u.Normal)', TEST_FACES, md);
B=gf_asm('generic', mim, 1, '(Interpolate(Grad_u,neighbour_elt).Normal)*(Interpolate(Grad_Test_u,neighbour_elt).Normal)', TEST_FACES, md);
err_v = err_v + norm(A-B);
if (err_v > 1E-13)
error('Test on neighbour element computation: error to big');
end
end
if (err > 2E-4)
error('Laplacian test: error to big');
end
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