.. _diffusion-laplace_coupling_lcbcs: diffusion/laplace_coupling_lcbcs.py =================================== **Description** Two Laplace equations with multiple linear combination constraints. The two equations are coupled by a periodic-like boundary condition constraint with a shift, given as a non-homogeneous linear combination boundary condition. Find :math:u such that: .. math:: \int_{\Omega_1} \nabla v_1 \cdot \nabla u_1 = 0 \;, \quad \forall v_1 \;, \int_{\Omega_2} \nabla v_2 \cdot \nabla u_2 = 0 \;, \quad \forall v_2 \;, u_1 = 0 \mbox{ on } \Gamma_{bottom} \;, u_2 = 1 \mbox{ on } \Gamma_{top} \;, u_1(\ul{x}) = u_2(\ul{x}) + a(\ul{x}) \mbox{ for } \ul{x} \in \Gamma = \bar\Omega_1 \cap \bar\Omega_2 u_1(\ul{x}) = u_1(\ul{y}) + b(\ul{y}) \mbox{ for } \ul{x} \in \Gamma_{left}, \ul{y} \in \Gamma_{right}, \ul{y} = P(\ul{x}) \;, u_1 = c_{11} \mbox{ in } \Omega_{m11} \subset \Omega_1 \;, u_1 = c_{12} \mbox{ in } \Omega_{m12} \subset \Omega_1 \;, u_2 = c_2 \mbox{ in } \Omega_{m2} \subset \Omega_2 \;, where :math:a(\ul{x}), :math:b(\ul{y}) are given functions (shifts), :math:P is the periodic coordinate mapping and :math:c_{11}, :math:c_{12} and :math:c_2 are unknown constant values - the unknown DOFs in :math:\Omega_{m11}, :math:\Omega_{m12} and :math:\Omega_{m2} are replaced by the integral mean values. View the results using:: \$ ./postproc.py square_quad.vtk -b --wireframe -d'u1,plot_warp_scalar,rel_scaling=1:u2,plot_warp_scalar,rel_scaling=1' .. image:: /../doc/images/gallery/diffusion-laplace_coupling_lcbcs.png :download:source code  .. literalinclude:: /../examples/diffusion/laplace_coupling_lcbcs.py