The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Abstract

This paper is devoted to the analysis of a linearized theta‐Galerkin finite element method for the time‐dependent coupled systems resulting from microsensor thermistor problems. Hereby, we focus on time discretization based on θ‐time stepping scheme with θ∈false[12,1false) including the standard Crank‐Nicolson ( θ=12) and the shifted Crank‐Nicolson ( θ=12+δ, where δ is the time‐step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in L2‐norm and the energy norm are established. For the fully discrete system, the optimal error estimates are derived for the standard Crank‐Nicolson, the shifted Crank‐Nicolson, and the general case where θ≠12+kδ with k=0,1 . Finally, numerical simulations that validate the theoretical findings are exhibited.

Existence of solution to a nonlocal conformable fractional thermistor problem

“…Nowadays, thermistors can be found everywhere, in airplanes, air conditioners, cars, computers, medical equipment, hair dryers, portable heaters, incubators, electrical outlets, refrigerators, digital thermostats, ice sensors and aircraft wings, ovens, stove tops and in all kinds of appliances. Knowing it, it is not a surprise that a great deal of attention is currently paid, by many authors, to the study of thermistor problems [28,29,30]. In [31], existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional-order derivatives were discussed.…”