%0 Journal Article 
%A Lahrache, Manar
%A Ortegón Gallego, Francisco
%A Rhoudaf, Mohamed
%T 3D numerical simulation of an anisotropic bead type thermistor and multiplicity of solutions
%D 2024
%@ 0378-4754
%U http://hdl.handle.net/10498/32742
%X We perform some 3D numerical experiments for the approximation of the solutions to a bead
type thermistor problem. We consider the case of a diagonal anisotropic diffusion matrix whose
𝑗�th entry is of the form|𝜕�𝑢�∕𝜕�𝑥�𝑗�|𝑝�𝑗�−2𝜕�𝑢�∕𝜕�𝑥�𝑗�,𝑢� being the temperature inside the thermistor and
the exponents 𝑝�𝑗�,1 ≤ 𝑗� ≤ 3, lie in the interval (1,+∞). We first show some existence results for
different notions of solutions, prove a maximum principle for each type of solution, and study
certain symmetry properties for these solutions in a bead type thermistor. These properties lead
us to the introduction of a symmetric solution and we show the existence of such a solution.
We have developed a numerical algorithm for the computation of the numerical solutions
in a bead type thermistor. This algorithm combines a fixed-point technique with a standard
finite element method (FEM). Some numerical tests have shown the existence of non-symmetric
solutions and this leads to multiple many solutions (at least three). We discuss the numerical
results obtained for different values of the exponents 𝑝�𝑗� and the applied voltage on different
meshes.
%K Thermistor problem
%K Anisotropic Sobolev space
%K Non-uniformly elliptic system
%K Finite element method
%K Numerical solution
%~ Universidad de Cádiz