A split least-squares characteristic mixed element method for nonlinear nonstationary convection–diffusion problem

Abstract

In this paper, a split least-squares characteristic mixed finite element method is proposed for solving nonlinear nonstationary convection–diffusion problem. By selecting the least-squares functional property, the resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes. The first sub-scheme is for the unknown variable u, which is the same as the standard characteristic Galerkin finite element approximation. The second sub-scheme is for the unknown flux σ. Theoretical analysis shows that the method yields the approximate solutions with optimal accuracy in L ²(Ω) norm for the primal unknown and in H(div; Ω) norm for the unknown flux, respectively. Some numerical examples are given to confirm our theory results.

Keywords:

• split least-squares algorithm
• method of characteristics
• mixed finite element
• nonlinear convection–diffusion problem
• convergence analysis

2000 AMS Subject Classifications :

• 65M60
• 65M25
• 65M15

Acknowledgements

The authors sincerely thank the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. This work was partially supported by the National Science Foundation for Young Scholars of China (11101431), the NSFC Tianyuan Mathematics Youth Fund (11126084), the Fundamental Research Funds for the Central Universities (12CX04082A,10CX04041A) and Shandong Province Natural Science Foundation of China (ZR2010AL020).