An error analysis of a finite element method for a system of nonlinear advection–diffusion–reaction equations

Liu, Biyue

doi:10.1016/j.apnum.2008.12.035

Cited by 12 publications

(7 citation statements)

References 19 publications

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“…In all those works, error estimates were established under certain time-step conditions. Moreover, linearized semi-implicit schemes have also been analyzed with certain time-step restrictions for many other nonlinear parabolic-type systems, such as Navier-Stokes equations [1,21,23,26,27], nonlinear thermistor problems [14,44], viscoelastic fluid flow [9,15,41], KdV equations [30], nonlinear Schrödinger equation [34,38] and some other equations [2,22,35]. A key issue in analysis of FEMs is the boundedness of the numerical solution in L ∞ norm or a stronger norm, which in a routine way can be estimated by the mathematical induction with an inverse inequality, such as,…”

Section: Introductionmentioning

confidence: 99%

Unconditional Convergence and Optimal Error Estimates of a Galerkin-Mixed FEM for Incompressible Miscible Flow in Porous Media

Sun

2013

SIAM J. Numer. Anal.

196

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In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal L 2 error estimates hold without any time-step (convergence) conditions, while all previous works require certain time-step restrictions. Our theoretical results provide a new understanding on commonly-used linearized schemes. The proof is based on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.

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Section: Introductionmentioning

confidence: 99%

Unconditional Convergence and Optimal Error Estimates of a Galerkin-Mixed FEM for Incompressible Miscible Flow in Porous Media

Sun

2013

SIAM J. Numer. Anal.

196

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“…Consider the example given in [22]. We take 1.287428e −5 9.125372e −4 3.569207e −5 1.602872e −2 Table 5.…”

Section: Examplementioning

confidence: 99%

“…At each time step, to enhance the accuracy of our numerical solution, we solve the discrete linear system (5.1) by using an iteration method. In Algorithm 1, the iteration procedure is detailed in steps (2.2)-(2.3), and the advantages in its implementation is described in [22]. Algorithm 1 :…”

Section: Implementation Detailsmentioning

confidence: 99%

Virtual element method for nonlinear convection–diffusion–reaction equation on polygonal meshes

Arrutselvi

Natarajan

2020

International Journal of Computer Mathematics

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In this work, we have discretized a system of time-dependent nonlinear convection-diffusionreaction equations with the virtual element method over the spatial domain and the Euler method for the temporal interval. For the nonlinear fully-discrete scheme, we prove the existence and uniqueness of the solution with Brouwer's fixed point theorem. To overcome the complexity of solving a nonlinear discrete system, we define an equivalent linear system of equations. A priori error estimate showing optimal order of convergence with respect to H 1 semi-norm was derived. Further, to solve the discrete system of equations, we propose an iteration method and a two-grid method. In the numerical section, the experimental results validate our theoretical estimates and point out the better performance of the two-grid method over the iteration method.

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“…Several scholars investigated Eq. (1.1) for example using an improved finite element approach [2], meshless local approaches [3,4], lattice Boltzmann technique [5], a front-tracking method [6], novel WENO methods [7], or a finite element method [8]. The interested readers can refer to [9,10] to get more information for Eq.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Legendre spectral element method (LSEM) for the two-dimensional system of a nonlinear stochastic advection-reaction-diffusion models

Abbaszadeh,

Khodadadian,

Dehghan

et al. 2019

Preprint

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In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection-reaction-diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank-Nicolson finite difference formulation. In the stochastic direction, we also employ a random variable W based on the Q−Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation. Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis.

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An error analysis of a finite element method for a system of nonlinear advection–diffusion–reaction equations

Cited by 12 publications

References 19 publications

Unconditional Convergence and Optimal Error Estimates of a Galerkin-Mixed FEM for Incompressible Miscible Flow in Porous Media

Unconditional Convergence and Optimal Error Estimates of a Galerkin-Mixed FEM for Incompressible Miscible Flow in Porous Media

Virtual element method for nonlinear convection–diffusion–reaction equation on polygonal meshes

Analysis of a Legendre spectral element method (LSEM) for the two-dimensional system of a nonlinear stochastic advection-reaction-diffusion models

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