Energy error estimates for a linear scheme to approximate nonlinear parabolic problems

Magenes, Enrico; Nochetto, Ricardo H.; Verdi, C.

doi:10.1051/m2an/1987210406551

Cited by 78 publications

(52 citation statements)

References 26 publications

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“…We notice that these two algorithms are equivalent; in fact, q n h in (4.18) is equal to We prove our stability result following [34]. First, we introduce some notation.…”

Section: Proof Of L 2 -Stabilitymentioning

confidence: 96%

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Cavalli¹,

Naldi²,

Perugia³

2012

SIAM J. Sci. Comput.

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Abstract. In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter ε. When ε → 0 + , the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing ε yields a consistent discretization of the original PDE. The numerical schemes obtained with this procedure do not require solving implicit nonlinear problems and possess the robustness of upwind discretizations. The proposed approximations are based on a discontinuous Galerkin method in space and on suitable implicitexplicit integration in time. Then, in principle, we can achieve any order of accuracy and obtain stable solutions, even when the diffusion equation becomes degenerate and solution singularities develop. Moreover, when needed, we can easily incorporate slope limiters within our schemes in order to handle spurious oscillatory phenomena. Some preliminary theoretical results are given, along with several numerical tests in one and two space dimensions, both for linear and nonlinear diffusion problems, including a degenerate diffusion equation, that provide numerical evidence of the properties of the presented approach. Key words. discontinuous Galerkin method, relaxation models, nonlinear diffusion AMS subject classifications. 65M60, 65M12, 35K55DOI. 10.1137/110827752 1. Introduction. Linear and nonlinear diffusion equations come from a variety of diffusion phenomena widely appearing in nature. They are suggested as mathematical models in many fields, such as filtration, phase transition, biochemistry, image analysis, and dynamics of biological groups. In the nonlinear case, the classical solutions to many of these PDEs fail to exist in finite time, even if the initial data are smooth. In such cases, suitable criteria have been introduced, which allow one to select physically relevant weak solutions beyond the singularity time.Recently, relaxation approximations to such PDEs have been introduced. These methods are based on replacing the equation by a semilinear hyperbolic system with stiff relaxation terms, tuned by a relaxation parameter ε. When ε → 0 + , the solution of this system "relaxes" onto the solution of the original PDE. Thus a consistent discretization of the relaxation system for ε = 0 yields a consistent discretization of the original PDE, as can be seen, for instance, in [29] and [2]. The advantage of this procedure is that the numerical scheme obtained in this fashion does not need approximate Riemann solvers for the convective term but possesses the robustness of upwind discretizations. Moreover, the complexity introduced by replacing the original

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“…We notice that these two algorithms are equivalent; in fact, q n h in (4.18) is equal to We prove our stability result following [34]. First, we introduce some notation.…”

Section: Proof Of L 2 -Stabilitymentioning

confidence: 96%

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Cavalli¹,

Naldi²,

Perugia³

2012

SIAM J. Sci. Comput.

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“…Magenes et al [3] considered a class of problems including the Stephan problem in the porous medium. During the calculation, they rest the strict equality by using the asymptotically correct Chernoff formulation.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Study of the Nonlinear Reaction-Diffusion Equation with Different Type of Absorbent Term by Homotopy Analysis Method

Gupta

Verma

2012

Zeitschrift Für Naturforschung A

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In this paper, based on the homotopy analysis method (HAM), a new powerful algorithm is used for the solution of the nonlinear reaction-diffusion equation. The algorithm presents the procedure of constructing a set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of the solution series by introducing an auxiliary parameter h. The solutions of the problem of presence and absence of absorbent term and external force for different particular cases are presented graphically.

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“…The numerical analysis of related problems was first considered in [4,16] in the case where b −1 is Lipschitz continuous and the equation written as a porous-medium equation. This scheme was adapted to more general porous medium equations in [10,13].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear elliptic-parabolic equations

Maitre

2002

ESAIM: M2AN

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Abstract. This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996;Halpern, 1967;Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995;Kačur, 1999).Mathematics Subject Classification. 65M12, 35K65, 35K55, 65N22.

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Energy error estimates for a linear scheme to approximate nonlinear parabolic problems

Cited by 78 publications

References 26 publications

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

A Numerical Study of the Nonlinear Reaction-Diffusion Equation with Different Type of Absorbent Term by Homotopy Analysis Method

Numerical analysis of nonlinear elliptic-parabolic equations

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