3D numerical simulation of an anisotropic bead type thermistor and multiplicity of solutions

Abstract

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.