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Adapting meshes and time-steps for phase change problems

Ricardo H. Nochetto, Alfred Schmidt, Claudio Verdi (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We address the numerical approximation of the two-phase Stefan problem and discuss an adaptive finite element method based on rigorous a posteriori error estimation and refinement/coarsening. We also investigate how to restrict coarsening for the resulting method to be stable and convergent. We review implementation issues associated with bisection and conclude with simulations of a persistent corner singularity, for which adaptivity is an essential tool.

Adaptive finite element methods for elliptic problems: Abstract framework and applications

Serge Nicaise, Sarah Cochez-Dhondt (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included into V. We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems...

Adaptive finite element relaxation schemes for hyperbolic conservation laws

Christos Arvanitis, Theodoros Katsaounis, Charalambos Makridakis (2001)

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

We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar...

Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws

Christos Arvanitis, Theodoros Katsaounis, Charalambos Makridakis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar...

Adaptive mesh refinement strategy for a non conservative transport problem

Benjamin Aymard, Frédérique Clément, Marie Postel (2014)

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

Long time simulations of transport equations raise computational challenges since they require both a large domain of calculation and sufficient accuracy. It is therefore advantageous, in terms of computational costs, to use a time varying adaptive mesh, with small cells in the region of interest and coarser cells where the solution is smooth. Biological models involving cell dynamics fall for instance within this framework and are often non conservative to account for cell division. In that case...

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Mehmet Ersoy, Frédéric Golay, Lyudmyla Yushchenko (2013)

Open Mathematics

We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to...

Adaptive Parameter Estimation of Hyperbolic Distributed Parameter Systems: Non-symmetric Damping and Slowly Time Varying Systems

H. T. Banks, M. A. Demetriou (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper a model reference-based adaptive parameter estimator for a wide class of hyperbolic distributed parameter systems is considered. The proposed state and parameter estimator can handle hyperbolic systems in which the damping sesquilinear form may not be symmetric (or even present) and a modification to the standard adaptive law is introduced to account for this lack of symmetry (or absence) in the damping form. In addition, the proposed scheme is modified for systems in which the input...

Adaptivity and variational stabilization for convection-diffusion equations

Albert Cohen, Wolfgang Dahmen, Gerrit Welper (2012)

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

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...

Adaptivity and variational stabilization for convection-diffusion equations∗

Albert Cohen, Wolfgang Dahmen, Gerrit Welper (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...

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