Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

Abstract

In this paper we study linearized backward differential formula (BDF) type schemes with Galerkin finite element approximations for the time-dependent nonlinear thermistor equations. Optimal L 2 error estimates for the proposed schemes are proved unconditionally. The proof consists of two steps. First, the boundedness of the numerical solution in certain strong norms is obtained by a temporal-spatial error splitting argument. Second, a traditional approach is used to provide an optimal L 2 error estimate for r -th order FEMs (r ≥ 1). Numerical experiments in both two and three dimensional spaces are conducted to confirm our theoretical analysis and show the high order accuracy and unconditional stability (convergence) of the linearized BDF-Galerkin FEMs.

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

“…Remark In order to bound B 5 , we estimate e n in H 1 ‐norm first and then derive e n in H 2 ‐norm. On the other hand, comparing with Liu and Gao, in order to achieve the temporal error with O ( τ 3 ) in H 2 ‐norm, we multiply by D τ e n and D τ Δ e n instead of 2 e n − e n − 1 and 2Δ e n − Δ e n − 1 , respectively. Further, we rewrite the error equation through to avoid transferring τ from one part in the inner product to the other hand.…”

“…The error is split into the temporal error u n − U n and the spatial error by a time‐discrete system with solution U n . Some original techniques, like rewriting the terms in the error equation and so on, are used to get higher order of the temporal error in H 2 ‐norm than that in Gao and Cai et al As it is shown in our paper, H 2 ‐error estimate of the temporal error is important for getting rid of the restriction of τ . It is also worth to point out that a much more general condition, where f ( u ) might not satisfy the globally condition or only locally Lipschitz continuous in u , is dealt with in the our work.…”

“…Different from Shi et al, Gao, and Cai et al, in this paper, we develop a linearized BDF‐3 type scheme for nonlinear Sobolev equation and discuss unconditional superconvergence estimates with bilinear element. Because the specularity of the BDF‐3 scheme, the particular inequality in Shi et al cannot work, and some skills, such as transferring τ from one part in the inner product to the other, in Shi et al and Shi and Wang cannot work either.…”

“…The linearized backward differential formula (BDF)–type schemes can achieve higher‐order accuracy without increasing the computation significantly. Recently, Gao discussed three‐step BDF (BDF‐3) type scheme for the time‐dependent nonlinear thermistor equations. Optimal error estimates were proved unconditionally through splitting technique.…”

Unconditional superconvergence analysis of a linearized Crank–Nicolson Galerkin FEM for generalized Ginzburg–Landau equation

“…However, due to the existence of nonlinearity, error analysis often requires some time-step grid ratio constraints, for example, τ = O(h d/2 ), d = 2, 3 in [3,11], τ = O(h) in [25] and τ = o(h 2 ) in [24], although numerical tests show the feasibility of the numerical methods for a large time step. To overcome this problem, an error splitting technique was proposed in [22], and then widely developed in [5,6,9,8,17,14,14,15,23].…”