Error analysis of finite element method for Poisson–Nernst–Planck equations

Self Cite

To improve the convergence rate in L2 norm from suboptimal to optimal for both electrostatic potential and ionic concentrations in Poisson‐Nernst‐Planck (PNP) system, we propose the mixed finite element method in this article to discretize the electrostatic potential equation, and still use the standard finite element method to discretize the time‐dependent ionic concentrations equations. Optimal error estimates in L∞ ([0,T];L2 ) norm for the electrostatic potential, and in L∞ ([0,T];L2 ) and L∞ ([0,T];H1 ) norms for the ionic concentrations are attained. As a by‐product, the electric field can also achieve a higher approximation order in contrast with the standard finite element method for PNP system. Numerical experiments are performed to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1924–1948, 2017

Section: Introductionmentioning

confidence: 99%

P^{2} P^{1}

element is used, the electrostatic potential is still approximated by the linear element, while its gradient, termed as the electric field, is approximated by the quadratic element.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of mixed finite element method for Poisson‐Nernst‐Planck system

Sun

2017

Self Cite

“…In this paper, we focus on the superconvergence analysis of bilinear finite element method (FEM) for the nonlinear Planck–Nernst–Poisson (PNP) equations , which describes the mass concentration of particles

p_{1}, p_{2} : (0, T] \to ℝ_{0}^{+}

and the electrostatic potential ψ : (0, T ] → ℝ, satisfying

\{\begin{matrix} left \\ p_{1 t} - \nabla \cdot (\nabla p_{1} + p_{1} \nabla ψ) = F_{1}, & (X, t) \in Ω \times J, \\ p_{2 t} - \nabla \cdot (\nabla p_{2} - p_{2} \nabla ψ) = F_{2}, & (X, t) \in Ω \times J, \\ - Δ ψ - p_{1} + p_{2} = F_{3}, & (X, t) \in Ω \times J, \end{matrix}

where Ω ⊂ ℝ 2 is a rectangular domain with its boundary ∂Ω ,

X = (x, y), J = (0, T], p_{italicit} = \frac{\partial p_{i}}{\partial t} (i = 1, 2)

and F i ( i = 1, 2, 3) are the reaction terms.…”

Section: Introductionmentioning

confidence: 99%

“…Let

(p_{1}^{0}, p_{2}^{0}, ψ^{0})

be the initial concentrations and potential. We employ the homogeneous Dirichlet boundary conditions as in , that is,

p_{1} = p_{2} = ψ = 0, (X, t) \in ∂Ω \times J .

…”

Section: Introductionmentioning

confidence: 99%

Superconvergence analysis of finite element method for Poisson–Nernst–Planck equations

Shi

Yang

2019

This article concerns with the superconvergence analysis of bilinear finite element method (FEM) for nonlinear Poisson–Nernst–Planck (PNP) equations. By employing high accuracy integral identities together with mean value technique, the superclose estimates in H1‐norm are derived for the semi‐discrete and the backward Euler fully‐discrete schemes, which improve the suboptimal error estimate in L2‐norm in the previous literature. Furthermore, the global superconvergence results in H1‐norm are obtained through interpolation postprocessing approach. Finally, a numerical example is provided to confirm the theoretical analysis.

“…To estimate B 2 , similar to the technique used in [28] (see (4.20) on p. 35), we give the following induction hypothesis and prove it by mathematical induction.…”

Section: Lemma 1 Assume That Umentioning

confidence: 99%

Superconvergent estimates of conforming finite element method for nonlinear time‐dependent Joule heating equations

Shi

Yang

2017

In this article, we study the superconvergence analysis of conforming bilinear finite element method (FEM) for nonlinear Joule heating equations. Based on the rigorous estimates together with high accuracy analysis of this element, mean value technique and interpolation postprocessing approach, the superclose and superconvergent estimates about the related variables in H 1 -norm are derived for semidiscrete and a linearized backward Euler fully discrete schemes, which extends the results of optimal estimates obtained for conforming FEMs in the previous literature. At last, a numerical experiment is performed to verify the theoretical analysis. K E Y W O R D SNonlinear Joule heating equations, bilinear FEM, semidiscrete and linearized fully discrete schemes, supercloseness and superconvergence analysis