On the Discretization of the Coupled Heat and Electrical Diffusion Problems [chapter]

Abdallah Bradji, Raphaèle Herbin
Numerical Methods and Applications  
We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear term which couples both equations. A finite element scheme and a finite volume scheme are considered for the discretization of the system; in both cases, we show that the approximate solution obtained with the scheme converges, up to a
more » ... up to a subsequence, to a solution of the coupled elliptic system. The discretization of such problems was undertaken in the last 10 years or so: convergence of the finite volume scheme was proven in [31] for the Laplace equation with right-hand-side measure; the proof was generalized in [18] to noncoercive convection diffusion problems. Convergence of the finite element scheme, with irregular data, on bi-dimensional polygonal domains was proven for Delaunay triangular meshes in [32, 28] and in [10] for three-dimensional tetrahedral meshes under
doi:10.1007/978-3-540-70942-8_1 dblp:conf/nma/BradjiH06 fatcat:wdq3arwsebhpnfprwguwnavhra