scholarly journals An Error Analysis for the Finite Element Approximation to the Steady-State Poisson-Nernst-Planck Equations

2013 ◽  
Vol 5 (1) ◽  
pp. 113-130 ◽  
Author(s):  
Ying Yang ◽  
Benzhuo Lu
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Nonlinear Partial Differential Equations ◽  
Planck Equation ◽  
Coupled System ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Numerical Examples ◽  
Delta Distribution ◽  
Biomolecular System

AbstractPoisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.

New Mixed Finite Element Formulations for Transient and Steady State Creep Problems

1989 ◽  
Vol 42 (11S) ◽  
pp. S150-S156
Author(s):  
Abimael F. D. Loula ◽  
Joa˜o Nisan C. Guerreiro
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Mixed Finite Element ◽  
Finite Element Approximation ◽  
Steady State Creep ◽  
Element Approximation ◽  
New Approach ◽  
Finite Element Approximations ◽  
Transient Problems ◽  
Transient And Steady State

We apply the mixed Petrov–Galerkin formulation to construct finite element approximations for transient and steady-state creep problems. With the new approach we recover stability, convergence, and accuracy of some Galerkin unstable approximations. We also present the main results on the numerical analysis and error estimates of the proposed finite element approximation for the steady problem, and discuss the asymptotic behavior of the continuum and discrete transient problems.

scholarly journals Finite element approximation of an incompressible chemically reacting non-Newtonian fluid

2018 ◽  
Vol 52 (2) ◽  
pp. 509-541 ◽  
Author(s):  
Seungchan Ko ◽  
Petra Pustějovská ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Newtonian Fluid ◽  
Power Law ◽  
Nonlinear Partial Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Partial Differential ◽  
Chemically Reacting

We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in the case of two space dimensions. Key technical tools include discrete counterparts of the Bogovskiĭ operator, De Giorgi’s regularity theorem in two dimensions, and the Acerbi–Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

scholarly journals Finite element approximation of steady flows of colloidal solutions

2021 ◽  
Author(s):  
Andrea Bonito ◽  
Vivette Girault ◽  
Diane Guignard ◽  
Kumbakonam R. Rajagopal ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Cauchy Stress ◽  
Partial Differential ◽  
Fixed Point Algorithm

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to  steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

On Finite Element Formulations for the Obstacle Problem – Mixed and Stabilised Methods

2017 ◽  
Vol 17 (3) ◽  
pp. 413-429 ◽  
Author(s):  
Tom Gustafsson ◽  
Rolf Stenberg ◽  
Juha Videman
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Numerical Examples ◽  
Finite Element Formulations

AbstractWe discuss the differences between the penalty, mixed and stabilised methods for the finite element approximation of the obstacle problem. The theoretical properties of the methods are discussed and illustrated through numerical examples.

scholarly journals Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

2021 ◽  
Author(s):  
Johnny Guzmán ◽  
Erik Burman
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Convection Term ◽  
Differentiation Formula ◽  
Numerical Examples ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Piecewise Affine ◽  
Time Discretisation

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection--diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank-Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the $\tau^2 + h^{p+{\frac12}}$ error estimates for the $L^2$-norm under either the standard hyperbolic CFL condition, when piecewise affine ($p=1$) approximation is used, or in the case of finite element approximation of order $p \ge 1$, a stronger, so-called $4/3$-CFL, i.e. $\tau \leq C h^{4/3}$. The theory is illustrated with some numerical examples.

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