Abstract
In this paper, a low order nonconforming mixed finite element method (MFEM) is studied with 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(h^{2}) for original variable u in broken H^{1} norm and the flux boldsymbol{p}=nabla u in (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(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^{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.
Highlights
Consider the following nonlinear reaction–diffusion equation:⎧ ⎪⎪⎨ut – u + f (u) = 0, (x, t) ∈ × (0, T],⎪⎪⎩uu((xx, t) 0) = = 0, u0(x, t) ∈ ∂ × (0, T], x∈, (1)where x = (x, y), 0 < T < ∞, and ∈ R2 is a rectangle with the boundary ∂ . u0 is a given function
We focus on mixed finite element method (MFEM) with EQr1ot element and zero order Raviart–Thomas (R–T) element for nonlinear reaction–diffusion equation (1) with reaction term f (u) = u3 – u
A striking feature of our analysis is that we control the Lp norm of φh ∈ M by broken H1 norm φh 1,h, which leads to the superclose and superconvergence unconditionally with nonconforming MFEM
Summary
A so-called a posteriori bound in L∞ norm was used to derive optimal error estimates of conventional (mixed) FEMs, time step size restrictions were required to circumvent the difficulties caused by nonlinear term f (u) [24]. Order O(h) result can be deduced (φh belongs to EQr1ot finite element space) Another novel splitting argument for consistency error is developed, by which the consistency errors are split into different parts and the time steps are transferred for one part of the integral to another. Remark 2 The key to the proof of (7) is the construction of a Lyapunov functional L(φ), which is not an easy thing for the general function f (·), and allows us to get rid of the assumption on boundedness of numerical solution uh 0,∞, which is usually necessary and inevitable in the traditional finite element analysis. Remark 3 With the help of (7) and (12), we derive the estimates for |(f (u)–f (uh), ξ )|, which are different from those in [24]
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