Finite element approximation to nonlinear coupled thermal problem

Abstract

a b s t r a c t A nonlinear coupled elliptic system modelling a large class of engineering problems was discussed in [A.F.D. Loula, J. Zhu, Finite element analysis of a coupled nonlinear system, Comp. Appl. Math. 20 (3) (2001) 321-339; J. Zhu, A.F.D. Loula, Mixed finite element analysis of a thermally nonlinear coupled problem, Numer. Methods Partial Differential Equations 22 (1) (2006) [180][181][182][183][184][185][186][187][188][189][190][191][192][193][194][195][196]. The convergence analysis of iterative finite element approximation to the solution was done under an assumption of 'small' solution or source data which guarantees the uniqueness of the nonlinear coupled system. Generally, a nonlinear system may have multiple solutions. In this work, the regularity of the weak solutions is further studied. The nonlinear finite element approximations to the nonsingular solutions are then proposed and analyzed. Finally, the optimal order error estimates in H 1 -norm and L 2 -norm as well as in W 1,p -norm and L p -norm are obtained.

Nonlinear scheme with high accuracy for nonlinear coupled parabolic–hyperbolic system

“…For example, a nonlinear scheme was studied in [18] for a two-temperature radiative diffusion problem, and it behaved more accurate than linear scheme with comparable costs. In [19], a nonlinear finite element approximation was studied for a coupled thermal problem expressed by elliptic system. In [20], it was pointed out that for traditional operator time-splitting methods, certain restriction on time step was needed due to stability requirement; while for nonlinear schemes, such restriction can be canceled, which means larger time steps are permissible without stability loss.…”

Numerical analysis and iteration acceleration of a fully implicit scheme for nonlinear diffusion problem with second‐order time evolution

“…For transient models with sharp diffusion coefficient and source varieties, nonlinear fully implicit (FI) discrete schemes are recommended to give more accurate solutions compared with linear schemes . Here, we mention nonlinear schemes for a two‐temperature radiative diffusion problem and a coupled thermal problem in and for example.…”