Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

Zhang, Bei; Chen, Shaochun; Zhao, Jikun

doi:10.1016/j.apnum.2015.02.002

Cited by 6 publications

(2 citation statements)

References 25 publications

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“…as an exact solution of Eq. (66). Similarly to Example 1, the functions f 1 , f 2 and f 3 on the RHS are determined by the exact solution.…”

Section: Examplementioning

confidence: 98%

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Adaptive finite element approximation for steady-state Poisson-Nernst-Planck equations

Hao¹,

Ma²,

Xu³

2020

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In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a key contribution, the steady-state Poisson-Nernst-Planck equations are studied systematically and rigorous analysis for a residual-based a posteriori error estimate of the nonlinear system is presented. With the help of Schauder fixed point theorem, we show the solution existence and uniqueness of the linearized system derived by taking G−derivatives of the nonlinear system, followed by the proof of the relationship between the error of solution and the a posteriori error estimator η. Numerical experiments are given to validate the efficiency of the a posteriori error estimator and demonstrate the expected rate of convergence. In the further tests, adaptive mesh refinements for geometrical singularities and boundary layer effects are successfully observed.

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“…as an exact solution of Eq. (66). Similarly to Example 1, the functions f 1 , f 2 and f 3 on the RHS are determined by the exact solution.…”

Section: Examplementioning

confidence: 98%

“…[2]. Besides, there are many studies on the a posterior estimation of linear singular perturbation problems with considering the boundary layer effects [1,16,13,3,66]. However, to solve cases with nonlinear and strongly coupling problems is challenging.…”

Section: Introductionmentioning

confidence: 99%