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 Dear Prof. Renard, thank you so much for your precious support. Just a few extra question:   Dear Domenico, Le 05/03/2015 00:17, Domenico Notaro a écrit : Dear GetFem Users, I am completely new of GetFem++ I am trying to implement a mixed formulation for the 3D Darcy problem. By simplifying terms useless for this issue, the weak formulation of the problem is   Find (u, p) in VxQ s.t.   (1)  (1/k u, v)   + (GRAD(p), v) = 0                  in \Omega     (2)  (u, GRAD(q)) - ((\alpha p, q)) + (f(p), q) = 0   in \Omega   where (.,.) and ((.,.)) indicate the L2 product on \Omega and \Gamma, respectively. [I integrate by part the divergence term because of the Robin BC: u.n = \alpha p  on \Gamma] I have one question for each of the following step: a) First of all, I tried to implement the assembly procedure for (GRAD(p), v). b) Then I tried to evaluate the satisfaction of the divergence constraint (||DIV(u)-f||), i.e. the strong equivalent of (2), for which I need to compute the divergence of a vector field. a) The implementation is the following, It seems to work properly - I have just a small doubt below - but I would like a double check from you expert users because this is my first implementation: /// Build the mixed pressure term /// $G = \int GRAD(p).v dx$ template void asm_darcy_G(MAT &                G,             const mesh_im  &    mim,             const mesh_fem &    mf_p,             const mesh_fem &    mf_u,             const mesh_region & rg = mesh_region::all_convexes()             ) {     GMM_ASSERT1(mf_p.get_qdim() == 1, "invalid data mesh fem (Qdim=1 required)");     GMM_ASSERT1(mf_u.get_qdim() > 1,  "invalid data mesh fem (Qdim=2,3 required)");     generic_assembly assem("M(#1,#2)+=comp(Grad(#1).vBase(#2))(:,i,:,i);");          assem.push_mi(mim);     assem.push_mf(mf_p);     assem.push_mf(mf_u);     assem.push_mat(G);     assem.assembly(rg); } /* end of asm_darcy_G */ (?) The output of this asm procedure is G^T not G, isn't it? (because A(i,j)=a(\phi_j,\phi_i) is a(.,.) is a non symmetric bilinear form) I mean, in this way I am assembling a mf_p.nb_dof() x mf_u.nb_dof() matrix while I need the transpose (to be multiplied then for the pressure vector P). Yes, this is correct, and this is indeed a mf_p.nb_dof() x mf_u.nb_dof() matrix. You can also use the high level generic assembly to perform this with an assembly string of the type "Grad_Test_p.Test2_u". OK, that's great! b) I tried to address this issue in two ways. b.1) By using the function getfem::compute_gradient - that seems to be the only way to compute derivatives - I computed the gradient tensor of the vector velocity and then extracted the divergence:     // Compute GRAD(U)     getfem::mesh_fem mf_gradU(mesh);     bgeot::pgeometric_trans pgt_t = bgeot::geometric_trans_descriptor(MESH_TYPE);     size_type N = pgt_t->dim();     mf_gradU.set_qdim(bgeot::dim_type(N), bgeot::dim_type(N)); //3x3     //mf_gradUt.set_classical_finite_element(0);     mf_gradUt.set_classical_discontinuous_finite_element(0);     vector_type gradU(mf_gradU.nb_dof());     getfem::compute_gradient(mf_U, mf_gradU, U, gradU);     //mf_U is at this level 'FEM_PK(3,1)'     // Compute DIV(U)     getfem::mesh_fem mf_Ui(mesh);     mf_Ui.set_classical_discontinuous_finite_element(0);     size_type nb_dof_Ui = mf_Ui.nb_dof(); //= mf_gradU.nb_dof()/(N*N)! NOT mf_U.nb_dof()/N     vector_type divU(nb_dof_Ui);              gmm::add(gmm::sub_vector(gradU, gmm::sub_interval(0*nb_dof_Ui, nb_dof_Ui)), divU);     gmm::add(gmm::sub_vector(gradU, gmm::sub_interval(4*nb_dof_Ui, nb_dof_Ui)), divU);     gmm::add(gmm::sub_vector(gradU, gmm::sub_interval(8*nb_dof_Ui, nb_dof_Ui)), divU); This seems to be incorrect. the component of the gradient are consecutives in the vector gradU. So you should use a gmm::sub_slice instead. OK, I will try to fix that. So, is the correct order   GRAD(U) = [DxUx,1  DyUx,1 DzUx,1  DxUx,2  DyUx,2  DzUx,2 ... ] or    GRAD(U) = [DxUx,1  DyUx,1 DzUx,1  DxUy,1  DyUy,1  DzUy,1 ... ] ? I mean, also the components are stored in this "sliced" way? And, in general, which approach do you think is better between the "algebraic" and the "assembly" ones?     // Compute ||DIV(U)-F|| by using asm_L2_norm (?) Here I assumed - but I am not sure at all - the function compute_gradient stores derivatives in the following order: GRAD(U) = [DxUx, DxUy, DxUz, DyUx, DyUy, ...] b.2) In order to feel more confident about the previous implementation I tried also to compute ||DIV(u)-f|| with an assembly approach: /// Compute the L2 norm of the residual of the divergence constraint /// $||DIV(u) - f|| = sqrt( \int (DIV(u) - f)^2 dx )$ template scalar_type asm_div_error_L2_norm( const VEC &U, const mesh_fem &mf_u,                        const VEC &F, const mesh_fem &mf_f,                        const mesh_im &mim,                        const mesh_region &rg = mesh_region::all_convexes() ) {     GMM_ASSERT1(mf_u.get_qdim() > 1, "invalid data mesh fem (Qdim>1 required)");     GMM_ASSERT1(mf_f.get_qdim() == 1, "invalid data mesh fem (Qdim=1 required)");     GMM_ASSERT1(U.size() == mf_u.nb_dof(), "invalid vector data (size=mf.nb_dof() required)");     GMM_ASSERT1(F.size() == mf_f.nb_dof(), "invalid vector data (size=mf.nb_dof() required)");     GMM_ASSERT1(U.size() == (F.size()*mf_u.get_qdim()), "invalid vector data (U.size=Qdim*F.size required)" );     generic_assembly     assem("u=data$1(#1);" "f=data$2(#2);"           "V()+=u(i).u(j).comp(vGrad(#1).vGrad(#1))(i,k,k,j,h,h)"               "-u(i).f(j).comp(vGrad(#1).Base(#2))(i,k,k,j)"               "-f(i).u(j).comp(Base(#2).vGrad(#1))(i,j,k,k)"               "+f(i).f(j).comp(Base(#2).Base(#2))(i,j);");     assem.push_mi(mim);     assem.push_mf(mf_u);     assem.push_mf(mf_f);     assem.push_data(U);     assem.push_data(F);     std::vector v(1);     assem.push_vec(v);     assem.assembly(rg);     return sqrt(v[0]); } (?) The results are quite different from those of b.1 (also if the order is the same) so I don't know what to trust - if at least one is correct! That's all! I am very sorry if it was too long and boring. You can also use here the high level generic assembly instead of the low-level one, this should be simpler. You can use the assembly string "sqr(Trace(Grad_u) - f)" OK, I will try that approach too! Apart from that, do you think this implementation should work? I am not sure about the index reduction: "V()+=u(i).u(j).comp(vGrad(#1).vGrad(#1))(i,k,k,j,h,h)". Yves. Many thanks. Best, Domenico user id: domenico_notaro --- Domenico Notaro mobile:  (+1) 412 983 0940 address: 3765 Childs Street, Pittsburgh PA 15213 mailto:  address@hidden Skype:   domenico.not