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

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 ∈ [ 1 2 , 1 ) including the standard Crank‐Nicolson ( θ = 1 2 ) and the shifted Crank‐Nicolson ( θ = 1 2 + δ , where δ is the time‐step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in L 2 ‐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 ≠ 1 2 + k δ with k =0,1 . Finally, numerical simulations that validate the theoretical findings are exhibited.