A new approach of superconvergence analysis of a low order nonconforming MFEM for reaction–diffusion equation

Boundary Value Problems

卷:2018

Houchao Zhang¹  An Wang¹ 

[1] School of Mathematics and Statistics, Pingdingshan University;

关键词: Reaction–diffusion equation;    Mixed finite element method;    Linearized fully discrete scheme;    Superconvergence;   

DOI  :  10.1186/s13661-018-1091-y

来源: DOAJ

【 摘 要 】

Abstract In this paper, a low order nonconforming mixed finite element method (MFEM) is studied with EQ1rot $EQ_{1}^{\mathrm{rot}}$ element and zero order Raviart–Thomas (R–T) element for a class of nonlinear reaction–diffusion equations. On the one hand, a priori bound is proved using Lyapunov functional, which leads to the global existence and uniqueness of the approximation solutions. Further, the superclose estimates of order O(h2) $O(h^{2})$ for original variable u in broken H1 $H^{1}$ norm and the flux p=∇u $\boldsymbol{p}=\nabla u$ in (L2)2 $(L^{2})^{2}$ norm are obtained respectively for a semi-discrete scheme. On the other hand, a linearized Crank–Nicolson fully-discrete scheme is established and the superclose estimates of order O(h2+τ2) $O(h^{2}+\tau^{2})$ are also obtained unconditionally by making full use of the special characters of the above mixed finite elements (MFEs) and two different splitting techniques, which are used to deal with the consistency errors and nonlinear terms, respectively. These approaches circumvent the restrictive condition on a time step size arising as an inverse inequality used to prove a posteriori bounds in L∞ $L^{\infty}$ norm, which is necessary for nonlinear problems for the conventional finite element analysis. What is more, the corresponding global superconvergent results are derived through interpolated postprocessing techniques. Finally, numerical results are provided to confirm the theoretical analysis. Here h is the subdivision parameter and τ is the time step.

【 授权许可】

Unknown