Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation (Journal of Computational Mathematics, 2018)
This paper develops a framework to deal with the unconditional superclose analysis ofnonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example,a new mixed finite element method (FEM) is established and the $τ$ -independent supercloseresults of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{→}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is anerror splitting technique, with which the time-discrete and the spatial-discrete systems areconstructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or requirecertain time step conditions. Finally, some numerical results are provided to confirm thetheoretical analysis, and show the efficiency of the proposed method.
Keywords: Nonlinear parabolic equation; Mixed FEM; Time-discrete and spatial-discrete systems; $τ$-independent superclose results.