Error Estimates of Optimal Order for Finite Element Methods with Interpolated Coefficients for the Nonlinear Heat Equation

Chen, Chuanmiao; Larsson, Stig; Zhang, Nai-Ying

doi:10.1093/imanum/9.4.507

Cited by 44 publications

(23 citation statements)

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“…There are also extensive literature in relation to applications of quasilinear parabolic equations to various physical and engineering problems, including numerical methods for this type of equations (cf. [1][2][3]7,11,14,17,19,20,22,[28][29][30]). The recent work [22] deals with the asymptotic behavior of the solution for a special type of parabolic operator which is motivated by some heat-transfer problems.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

Pao¹,

Ruan²

2007

Journal of Mathematical Analysis and Applications

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The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i (u i ) may have the property D i (0) = 0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a twocomponent competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.

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Section: Introductionmentioning

confidence: 99%

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

Pao¹,

Ruan²

2007

Journal of Mathematical Analysis and Applications

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“…The use of product approximation of the coefficients in Galerkin finite element methods for nonlinear problems has been analyzed by several authors (e.g., [8,28,7,12,17]). They applied the interpolation operator to the nonlinear coefficients and obtained error estimates of optimal convergence order with much less computational effort.…”

Section: Introductionmentioning

confidence: 99%

Two-Scale Product Approximation for Semilinear Parabolic Problems in Mixed Methods

Kim

Park²,

Seo

2014

Journal of the Korean Mathematical Society

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Abstract. We propose and analyze two-scale product approximation for semilinear heat equations in the mixed finite element method. In order to efficiently resolve nonlinear algebraic equations resulting from the mixed method for semilinear parabolic problems, we treat the nonlinear terms using some interpolation operator and exploit a two-scale grid algorithm. With this scheme, the nonlinear problem is reduced to a linear problem on a fine scale mesh without losing overall accuracy of the final system. We derive optimal order L ∞ ((0, T ]; L 2 (Ω))-error estimates for the relevant variables. Numerical results are presented to support the theory developed in this paper.

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“…In 1980, this method was introduced and analyzed firstly for the semilinear parabolic problems by Zlamal [46]. Later, Larson, Thomee and Zhang [22] studied the semidiscrete linear triangular FEM with interpolated coefficients, and Chen, Larson and Zhang [8] derived almost optimal order convergence on a uniform triangular mesh by using the piecewise linear finite element space and superconvergence techniques. Xiong and Chen [37][38][39] studied the superconvergence of triangular quadratic FEM for the nonlinear ordinary differential equation and the Q 1 -conforming rectangular FEM with interpolated coefficients for the semilinear elliptic problem, respectively.…”

Section: Introductionmentioning

confidence: 99%

P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations

Feng

et al. 2011

J Sci Comput

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In this paper we use P 1 -nonconforming quadrilateral finite volume methods with interpolated coefficients to solve the semilinear elliptic problems. Two types of control volumes are applied. Optimal error estimates in H 1 -norm on the quadrilateral mesh and superconvergence of derivative on the rectangular mesh are derived by using the continuity argument, respectively. In addition, numerical experiments are presented adequately to confirm the theoretical analysis and optimal error estimates in L 2 -norm is also observed obviously. Compared with the standard Q 1 -conforming quadrilateral element, numerical results of the proposed finite volume methods show its better performance than others.Keywords Semilinear elliptic equations · Finite volume methods · P 1 -nonconforming quadrilateral element · Q 1 -conforming quadrilateral element · Superconvergence

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Error Estimates of Optimal Order for Finite Element Methods with Interpolated Coefficients for the Nonlinear Heat Equation

Cited by 44 publications

References 0 publications

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

Two-Scale Product Approximation for Semilinear Parabolic Problems in Mixed Methods

P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations

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