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A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

Abigail Wacher (2013)

Open Mathematics

We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both...

A domain splitting method for heat conduction problems in composite materials

Friedrich Karl Hebeker (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a domain decomposition method for some unsteady heat conduction problem in composite structures. This linear model problem is obtained by homogenization of thin layers of fibres embedded into some standard material. For ease of presentation we consider the case of two space dimensions only. The set of finite element equations obtained by the backward Euler scheme is parallelized in a problem-oriented fashion by some noniterative overlapping domain splitting method, eventually enhanced...

A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation

J. Zhu, Y.-T. Zhang, S. A. Newman, M. S. Alber (2009)

Mathematical Modelling of Natural Phenomena

Skeletal patterning in the vertebrate limb, i.e., the spatiotemporal regulation of cartilage differentiation (chondrogenesis) during embryogenesis and regeneration, is one of the best studied examples of a multicellular developmental process. Recently [Alber et al., The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb, Bulletin of Mathematical Biology, 2008, v70, pp. 460-483], a simplified two-equation reaction-diffusion system was developed to describe the interaction...

A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme

Hyam Abboud, Toni Sayah (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH...

A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems – Analysis, assessments and applications to parameter estimation

D. Chapelle, A. Gariah, P. Moireau, J. Sainte-Marie (2013)

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

We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder – in the construction procedure – and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the...

A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model

David J. Knezevic, Endre Süli (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We examine a heterogeneous alternating-direction method for the approximate solution of the FENE Fokker–Planck equation from polymer fluid dynamics and we use this method to solve a coupled (macro-micro) Navier–Stokes–Fokker–Planck system for dilute polymeric fluids. In this context the Fokker–Planck equation is posed on a high-dimensional domain and is therefore challenging from a computational point of view. The heterogeneous alternating-direction scheme combines a spectral Galerkin method for...

A hyperbolic model for convection-diffusion transport problems in CFD: numerical analysis and applications.

Héctor Gómez, Ignasi Colominas, Fermín L. Navarrina, Manuel Casteleiro (2008)

RACSAM

In this paper we present a numerical study of the hyperbolic model for convection-diffusion transport problems that has been recently proposed by the authors. This model avoids the infinite speed paradox, inherent to the standard parabolic model and introduces a new parameter called relaxation time. This parameter plays the role of an “inertia” for the movement of the pollutant. The analysis presented herein is twofold: first, we perform an accurate study of the 1D steady-state equations and its...

A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal model

F. M. Guillén-González, J. V. Gutiérrez-Santacreu (2013)

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

In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen–Leslie nematic liquid crystal model by means of a Ginzburg–Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen–Leslie problem...

A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations

Gabriel R. Barrenechea, Volker John, Petr Knobloch (2013)

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

An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived...

A Multiscale Model Reduction Method for Partial Differential Equations

Maolin Ci, Thomas Y. Hou, Zuoqiang Shi (2014)

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

We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show...

A “Natural” Norm for the Method of Characteristics Using Discontinuous Finite Elements : 2D and 3D case

Jacques Baranger, Ahmed Machmoum (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the numerical approximation of a first order stationary hyperbolic equation by the method of characteristics with pseudo time step k using discontinuous finite elements on a mesh 𝒯 h . For this method, we exhibit a “natural” norm || ||h,k for which we show that the discrete variational problem P h k is well posed and we obtain an error estimate. We show that when k goes to zero problem ( P h k ) (resp. the || ||h,k norm) has as a limit problem (Ph) (resp. the || ||h norm) associated to the...

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