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The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems
Author(s) -
Mbehou Mohamed
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4678
Subject(s) - crank–nicolson method , discretization , mathematics , galerkin method , finite element method , norm (philosophy) , thermistor , time stepping , mathematical analysis , physics , quantum mechanics , political science , law , thermodynamics
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.