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A mixed formulation of a sharp interface model of stokes flow with moving contact lines

Shawn W. Walker (2014)

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

Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking...

A model of macroscale deformation and microvibration in skeletal muscle tissue

Bernd Simeon, Radu Serban, Linda R. Petzold (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper deals with modeling the passive behavior of skeletal muscle tissue including certain microvibrations at the cell level. Our approach combines a continuum mechanics model with large deformation and incompressibility at the macroscale with chains of coupled nonlinear oscillators. The model verifies that an externally applied vibration at the appropriate frequency is able to synchronize microvibrations in skeletal muscle cells. From the numerical analysis point of view, one faces...

A modified Cayley transform for the discretized Navier-Stokes equations

K. A. Cliffe, T. J. Garratt, Alastair Spence (1993)

Applications of Mathematics

This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the...

A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction

Roland Ernst, Bernd Flemisch, Barbara Wohlmuth (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

A new Schwarz method for nonlinear systems is presented, constituting the multiplicative variant of a straightforward additive scheme. Local convergence can be guaranteed under suitable assumptions. The scheme is applied to nonlinear acoustic-structure interaction problems. Numerical examples validate the theoretical results. Further improvements are discussed by means of introducing overlapping subdomains and employing an inexact strategy for the local solvers.

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...

A new conservative finite difference scheme for Boussinesq paradigm equation

Natalia Kolkovska, Milena Dimova (2012)

Open Mathematics

A family of nonlinear conservative finite difference schemes for the multidimensional Boussinesq Paradigm Equation is considered. A second order of convergence and a preservation of the discrete energy for this approach are proved. Existence and boundedness of the discrete solution on an appropriate time interval are established. The schemes have been numerically tested on the models of the propagation of a soliton and the interaction of two solitons. The numerical experiments demonstrate that the...

A new domain decomposition method for the compressible Euler equations

Victorita Dolean, Frédéric Nataf (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf, C. R. Acad. Sci. Paris Sér. I325 (1997) 1211–1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains,...

A new energy conservative scheme for regularized long wave equation

Yuesheng Luo, Ruixue Xing, Xiaole Li (2021)

Applications of Mathematics

An energy conservative scheme is proposed for the regularized long wave (RLW) equation. The integral method with variational limit is used to discretize the spatial derivative and the finite difference method is used to discretize the time derivative. The energy conservation of the scheme and existence of the numerical solution are proved. The convergence of the order O ( h 2 + τ 2 ) and unconditional stability are also derived. Numerical examples are carried out to verify the correctness of the theoretical analysis....

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

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 , ∞ ( ℒ 2 ) is proved. An ℒ ∞ ( ℋ 1 ) -error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations...

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