diff --git a/chapter3/robin_neumann_dirichlet.ipynb b/chapter3/robin_neumann_dirichlet.ipynb index ce17f02c..bbeddfb6 100644 --- a/chapter3/robin_neumann_dirichlet.ipynb +++ b/chapter3/robin_neumann_dirichlet.ipynb @@ -49,13 +49,13 @@ "On the Dirichlet part ($\\Gamma_D^i$), the boundary integral vanishes since $v=0$. On the remaining part of the boundary, we split the boundary into contributions from the Neumann parts ($\\Gamma_N^i$) and Robin parts ($\\Gamma_R^i$). Inserting the boundary conditions, we obtain\n", "\n", "$$\n", - "-\\int_{\\Omega}\\kappa\\frac{\\partial u }{\\partial n }v~\\mathrm{d} s=\\sum_i \\int_{\\Gamma_N^i} g_i~\\mathrm{d} s + \\sum_i\\int_{\\Gamma_R^i}r_i(u-s_i)~\\mathrm{d}s.\n", + "-\\int_{\\Omega}\\kappa\\frac{\\partial u }{\\partial n }v~\\mathrm{d} s=\\sum_i \\int_{\\Gamma_N^i} g_i v~\\mathrm{d} s + \\sum_i\\int_{\\Gamma_R^i}r_i(u-s_i)v~\\mathrm{d}s.\n", "$$\n", "\n", "Thus we have the following variational problem\n", "\n", "$$\n", - "F(u, v)=\\int_\\Omega \\kappa \\nabla u \\cdot \\nabla v~\\mathrm{d} x + \\sum_i\\int_{\\Gamma_N^i}g_i v~\\mathrm{d}s +\\sum_i\\int_{\\Gamma_R^i}r_i(u-s_i)~\\mathrm{d}s - \\int_\\Omega fv~\\mathrm{d} x = 0.\n", + "F(u, v)=\\int_\\Omega \\kappa \\nabla u \\cdot \\nabla v~\\mathrm{d} x + \\sum_i\\int_{\\Gamma_N^i}g_i v~\\mathrm{d}s +\\sum_i\\int_{\\Gamma_R^i}r_i(u-s_i)v~\\mathrm{d}s - \\int_\\Omega fv~\\mathrm{d} x = 0.\n", "$$\n", "\n", "We have been used to writing the variational formulation as $a(u,v)=L(v)$, which requires that we identify the integrals dependent on the trial function $u$ and collect these in $a(u,v)$, while the remaining terms form $L(v)$. We note that the Robin condition has a contribution to both $a(u,v)$ and $L(v)$. \n", @@ -396,7 +396,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.10.12" + "version": "3.12.2" } }, "nbformat": 4,