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

He, Mingyan; Sun, Pengtao

doi:10.1002/num.22170

Cited by 14 publications

(11 citation statements)

References 51 publications

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“…Remark 1 For Galerkin FEMs, [24] only deduced the suboptimal error estimates in L 2 -norm with linear element, and [5] improved it to optimal by mixed FEM with Taylor-Hood-type elements. But when this method is applied to deal with problem (1) in 3D, the approximating function of FE space for p i must be more than quadratic polynomial instead of the linear one as that of .…”

Section: Theorem 42 Under the Conditions Of Theorem 41 We Have Formentioning

confidence: 99%

Section: Theorem 42 Under the Conditions Of Theorem 41 We Have Formentioning

confidence: 99%

“…Similar to the semidiscrete case, we can apply the interpolation postprocessing operator I 2 h to get the following global superconvergence estimate.Theorem Under the conditions of Theorem 4.1 , we have for 1 ≤ n ≤ N that

‖ p_{i}^{n} - I_{2 h} p_{italicih}^{n} ‖_{h} + ‖ ψ^{n} - I_{2 h} ψ_{h}^{n} ‖_{h} \leq C (h^{2} + τ), i = 1, 2 .

Remark For Galerkin FEMs, [24] only deduced the suboptimal error estimates in L 2 ‐norm with linear element, and [5] improved it to optimal by mixed FEM with Taylor–Hood‐type elements. But when this method is applied to deal with problem (1) in 3D, the approximating function of FE space for p i must be more than quadratic polynomial instead of the linear one as that of ψ .…”

Section: Superconvergence Analysis Of Backward‐euler Fully Discrete Smentioning

confidence: 99%

“…But when this method is applied to deal with problem (1) in 3D, the approximating function of FE space for p i must be more than quadratic polynomial instead of the linear one as that of ψ . The main reason is the restriction of the mathematics induction assumption used in [5]. Now, the above two defects are removed for that we not only derive the superconvergent results but also can extend them to 3D case easily. Remark One can check that the regularity assumptions of p i ( i = 1, 2) and ψ for our superconvegence analysis are weaker than that of [5, 24] for the corresponding convergence analysis, and also weaker than that of [17] for the superconvegence analysis of bilinear element. Remark The results obtained herein are also valid to the boundary conditions

\frac{\partial p_{1}}{\partial n} = \frac{\partial p_{2}}{\partial n} = \frac{\partial ψ}{\partial n} = 0

, if

V_{h}^{0}

is replaced by V h and the proofs are slightly modified.…”

Section: Superconvergence Analysis Of Backward‐euler Fully Discrete Smentioning

confidence: 99%

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Nonconforming finite element method for coupled Poisson–Nernst–Planck equations

Shi

2021

Numerical Methods Partial

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A nonconforming finite element method (FEM) is developed and investigated for the coupled Poisson-Nernst-Planck (PNP) equations with low order EQ rot 1 element. Then, by use of the special properties of this element, that is, the interpolation operator is equivalent to its projection operator, and the consistency error estimate can reach order of O(h 2) which is one order higher than that of its interpolation error estimates when the exact solution belongs to H 3 (Ω), the superclose estimates of order O(h 2) and O(h 2 + τ) in the broken H 1-norm are derived with new techniques for the semidiscrete scheme and backward Euler fully discrete scheme, respectively. Further, through employing interpolation postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some numerical results are provided to confirm the theoretical analysis. It seems that our results have never been found in the existing literature. Here h and τ denote the mesh size and time step, respectively.

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Section: Theorem 42 Under the Conditions Of Theorem 41 We Have Formentioning

confidence: 99%

Section: Theorem 42 Under the Conditions Of Theorem 41 We Have Formentioning

confidence: 99%

‖ p_{i}^{n} - I_{2 h} p_{italicih}^{n} ‖_{h} + ‖ ψ^{n} - I_{2 h} ψ_{h}^{n} ‖_{h} \leq C (h^{2} + τ), i = 1, 2 .

Section: Superconvergence Analysis Of Backward‐euler Fully Discrete Smentioning

confidence: 99%

\frac{\partial p_{1}}{\partial n} = \frac{\partial p_{2}}{\partial n} = \frac{\partial ψ}{\partial n} = 0

, if

V_{h}^{0}

is replaced by V h and the proofs are slightly modified.…”

Section: Superconvergence Analysis Of Backward‐euler Fully Discrete Smentioning

confidence: 99%

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Nonconforming finite element method for coupled Poisson–Nernst–Planck equations

Shi

2021

Numerical Methods Partial

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“…Computational studies started in the 1960s [28,38] and many discretization methods have been used for the drift-diffusion system in the past decades. For an extensive body of literature devoted to this subject we refer to, e.g., the finite difference method [30,39,50,55], the finite volume method [3,4,[11][12][13], the standard finite element method (FEM) [35,52,62], and mixed FEM [36,40]. Furthermore, there are many new models in which the drift-diffusion equation coupled with other PDEs; such as Stokes [41], Navier-Stokes [61] and Darcy flow [31].…”

Section: Introductionmentioning

confidence: 99%

An HDG Method for the Time-dependent Drift–Diffusion Model of Semiconductor Devices

2019

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We propose a hybridizable discontinuous Galerkin (HDG) finite element method to approximate the solution of the time dependent drift-diffusion problem. This system involves a nonlinear convection diffusion equation for the electron concentration u coupled to a linear Poisson problem for the the electric potential φ. The non-linearity in this system is the product of the ∇φ with u. An improper choice of a numerical scheme can reduce the convergence rate. To obtain optimal HDG error estimates for φ, u and their gradients, we utilize two different HDG schemes to discretize the nonlinear convection diffusion equation and the Poisson equation. We prove optimal order error estimates for the semidiscrete problem. We also present numerical experiments to support our theoretical results. the above model, for example including additional particle transport equations (for example, for holes) and recombination terms. However the above system contains the principle difficulty from the point of view of proving convergence: the term ∇ · (u∇φ).Theoretical and numerical studies for this type of partial differential equation (PDE) have a long history. For the theoretical analysis of the drift-diffusion system; see [5,6,33,34,46,56] and the references therein. Computational studies started in the 1960s [28,38] and many discretization methods have been used for the drift-diffusion system in the past decades. For an extensive body of literature devoted to this subject we refer to, e.g., the finite difference method [30,39,50,55], the finite volume method [3,4,[11][12][13], the standard finite element method (FEM) [35,52,62], and mixed FEM [36,40]. Furthermore, there are many new models in which the drift-diffusion equation coupled with other PDEs; such as Stokes [41], Navier-Stokes [61] and Darcy flow [31]. However these extensions are outside the scope of this paper.The product of the gradient of the electric potential, ∇φ with electron concentration u in (1.1a) can cause a reduction in the convergence rate of the solution if the numerical schemes for the two equations are not properly devised. In [62], they obtained an optimal convergence rate in H 1 norm but a suboptimal in L 2 norm by the standard FEM. To overcome the convergence order reduction, a new method was proposed to discretize the system (1.1); mixed FEM for Poisson equation (1.1b) and standard FEM for (1.1a),. This scheme provides optimal error estimates for u and φ in both H 1 or H(div) as appropriate as well as in the L 2 norm. Very recently, the authors in [36] obtained an optimal convergence rate by using mixed FEM for both (1.1a) and (1.1b).In the drift-diffusion model, typically, the magnitude of ∇φ is huge (see [8]). Therefore, it is natural to consider the discontinuous Galerkin (DG) method to discretize the system (1.1). In [51], a local DG (LDG) method was used to study a 1D drift-diffsuion equation, they obtained an optimal convergence rate by using an important relationship between the gradient and interface jump of the numerical solution with the independen...

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Unconditional superconvergence analysis of a structure-preserving finite element method for the Poisson-Nernst-Planck equations

Yang,

2024

Adv Comput Math

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

Cited by 14 publications

References 51 publications

Nonconforming finite element method for coupled Poisson–Nernst–Planck equations

Nonconforming finite element method for coupled Poisson–Nernst–Planck equations

An HDG Method for the Time-dependent Drift–Diffusion Model of Semiconductor Devices

Unconditional superconvergence analysis of a structure-preserving finite element method for the Poisson-Nernst-Planck equations

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