Stability and convergence of two discrete schemes  for a degenerate solutal 
non-isothermal  phase-field model

Francisco Guillén-González; Juan Vicente Gutiérrez-Santacreu

Displaying similar documents to “Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model”

Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes

Abdallah Bradji, Jürgen Fuhrmann (2013)

Applications of Mathematics

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A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the...

A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

Mario Ohlberger (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_{t} + \nabla \cdot (𝐮 f (c)) - \nabla \cdot (D \nabla c) + λ c = 0$ . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the $L^{1}$ -norm, independent of the diffusion parameter $D$ . The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability...

Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

Konstantinos Chrysafinos (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at...

A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method

Abdallah Bradji, Jürgen Fuhrmann (2014)

Mathematica Bohemica

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Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of $𝒲^{1, \infty} (ℒ^{2})$ is proved. An $ℒ^{\infty} (ℋ^{1})$ -error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations...

Skipping transition conditions in error estimates for finite element discretizations of parabolic equations

Stefano Berrone (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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In this paper we derive error estimates for the heat equation. The time discretization strategy is based on a -method and the mesh used for each time-slab is independent of the mesh used for the previous time-slab. The novelty of this paper is an upper bound for the error caused by the coarsening of the mesh used for computing the solution in the previous time-slab. The technique applied for deriving this upper bound is independent of the problem and can be generalized to other time...

Error estimates for finite volume scheme for Perona--Malik equation.

Handlovičová, A., Krivá, Z. (2005)

Acta Mathematica Universitatis Comenianae. New Series

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Lagrangian and moving mesh methods for the convection diffusion equation

Konstantinos Chrysafinos, Noel J. Walkington (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

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We propose and analyze a semi Lagrangian method for the convection-diffusion equation. Error estimates for both semi and fully discrete finite element approximations are obtained for convection dominated flows. The estimates are posed in terms of the projections constructed in [Chrysafinos and Walkington, (2006) 2478–2499; Chrysafinos and Walkington, (2006) 349–366] and the dependence of various constants upon the diffusion parameter is characterized. Error estimates independent...

Some new error estimates for finite element methods for second order hyperbolic equations using the Newmark method

Abdallah Bradji, Jürgen Fuhrmann (2014)

Mathematica Bohemica

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We consider a family of conforming finite element schemes with piecewise polynomial space of degree $k$ in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is $h^{k} + τ^{2}$ in the discrete norms of $ℒ^{\infty} (0, T; ℋ^{1} (Ω))$ and $𝒲^{1, \infty} (0, T; ℒ^{2} (Ω))$ , where $h$ and $τ$ are the mesh size of the spatial and temporal discretization,...