Euler Scheme In Time Research Articles

Space-Time Residual-Based <italic>a posteriori</italic> Estimator for the <inline-formula> <tex-math notation="LaTeX">$A-\varphi$ </tex-math></inline-formula> Formulation in Eddy Current Problems

In this paper, an a posteriori residual error estimator is presented for the 3-D eddy current problem modeled by the space-time ${A}-\varphi $ potential formulation. It is solved by the finite element method in space and the backward Euler scheme in time. Once the reliability as well as the efficiency of the estimator is established, two numerical tests are proposed: 1) an analytical one to validate the theoretical results and 2) a physical one to illustrate the performance for a real eddy current problem.

Quasi Wavelet based numerical method for Volterra integro-differential equations on unbounded spatial domains

In this paper we study the numerical solution of the parabolic Volterra integro-differential equation with a memory term on one-dimensional unbounded domain. A new numerical method is proposed by applying the forward Euler scheme in time and the quasi-wavelet based numerical scheme in space. Some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.

Stabilized finite element discretization applied to an operator-splitting method of population balance equations

An operator-splitting method is applied to transform the population balance equation into two subproblems: a transient transport problem with pure advection and a time-dependent convection–diffusion problem. For discretizing the two subproblems the discontinuous Galerkin method and the streamline upwind Petrov–Galerkin method combined with a backward Euler scheme in time are considered. Standard energy arguments lead to error estimates with a lower bound on the time step length. The stabilization vanishes in the time-continuous limit case. For this reason, we follow a new technique proposed by John and Novo for transient convection–diffusion–reaction equations and extend it to the case of population balance equations. We also compare numerically the streamline upwind Petrov–Galerkin method and the local projection stabilization method.

Convergence of a discrete scheme in a regularization method for the quasi-steady Maxwell system in a nonhomogeneous conducting medium

Convergence of the discrete solution to the solution of a regularized system of the Maxwell equations written in terms of a vector magnetic potential with a special calibration of medium’s conduction is considered. The problem is discretized by a Nedelec vector finite element method in space and by an implicit Euler scheme in time. An optimal theoretical energy estimate of the approximate solution error in 3D Lipschitz polyhedral domains is obtained.

Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystals model

A linear fully discrete mixed scheme, using C 0 finite elements in space and a semi-implicit Euler scheme in time, is considered for solving a penalized nematic liquid crystal model (of the Ginzburg-Landau type). We prove: 1) unconditional stability and convergence towards weak solutions, and 2) first-order optimal error estimates for regular solutions (but without imposing the well-known global compatibility condition for the initial pressure in the Navier-Stokes framework). These results are valid in a general connected polygon or in a Lipschitz polyhedral domain (without any constraints on its angles). Finally, since the scheme couples the unknowns, we propose several algorithms for decoupling the computation of these unknowns and establish their rates of convergence in convex domains when the mesh size is sufficiently small compared to the time step.

A posteriori error estimators for the fully discrete time dependent Stokes problem with some different boundary conditions

In this paper we study the time dependent Stokes problem with some different boundary conditions. We establish a decoupled variational formulation into a system of velocity and a Poisson equation for the pressure. Hence, the velocity is approximated with curl conforming finite elements in space and Euler scheme in time and the pressure with standard continuous elements in space and Euler scheme in time. Finally, we establish optimal a priori and a posteriori estimates.

On the convergence of the wavelet-Galerkin method for nonlinear filtering

On the convergence of the wavelet-Galerkin method for nonlinear filteringThe aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.

A posteriori error estimate for a one-dimensional pollution problem in porous media

We are interested in the discretization of a time-dependent pollution problem modeling the mass transfer of contaminant in porous media, by the implicit Euler scheme in time and vertex-centered finite volumes in space. The error estimator consists of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization. To cite this article: R. Aboulaich et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation

We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, various finite volume, and mixed finite element methods in space and the backward Euler scheme in time. Extensions to conforming and nonconforming finite element spatial discretizations are also outlined. Our estimates are based on a $H^1$-conforming reconstruction of the potential, continuous and piecewise affine in time, and a locally conservative $\mathbf{H}(\mathrm{div})$-conforming reconstruction of the flux, piecewise constant in time. They yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative. Local-in-time lower bounds are also derived; for nonconforming methods on time-varying meshes, the lower bounds require a mild parabolic-type constraint on the meshsize.

TIME AND SPACE ADAPTIVITY FOR THE SECOND-ORDER WAVE EQUATION

The aim of this paper is to show that, for a linear second-order hyperbolic equation discretized by the backward Euler scheme in time and continuous piecewise affine finite elements in space, the adaptation of the time steps can be combined with spatial mesh adaptivity in an optimal way. We derive a priori and a posteriori error estimates which admit, as much as it is possible, the decoupling of the errors committed in the temporal and spatial discretizations.

A posteriorierror analysis of the fully discretized time-dependent Stokes equations

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

Time-Space Discretization of the Nonlinear Hyperbolic Systemutt= div (\sigma(Du)+Dut)

The numerical treatment of the hyperbolic system of nonlinear wave equations with linear viscosity, $u_{tt}= {\rm div}(\sigma(Du)+Du_t)$, is studied for a large class of globally Lipschitz continuous functions $\sigma$, including nonmonotone stress-strain relations. The analyzed method combines an implicit Euler scheme in time with Courant (continuous and piecewise affine) finite elements in space for a class of varying time steps with varying meshes. Explicit a priori error bounds in $L^\infty(L^2)$, $L^2(W^{1,2})$, and W1,2 (L2 ) are established for the solutions of the fully discrete scheme.

The mortar edge element method on non‐matching grids for eddy current calculations in moving structures

AbstractThe subject presented in this paper concerns the approximation of the eddy current problem in non‐stationary geometries with sliding interfaces. The physical system is supposed to be composed of two solid parts: a fixed one (stator) and a moving one (rotor) which slides in contact with the stator. We consider a two‐dimensional mathematical model based on the transverse electric formulation of the eddy currents problem in the time domain and the primary unknown is the electric field vector. The first‐order approximation of the problem that we propose here is based on the mortar element method combined with the edge element discretization in space and an implicit Euler scheme in time. Numerical results illustrate the accuracy of the method and allow to understand the influence of the rotor movement on the currents distribution. Copyright © 2001 John Wiley & Sons, Ltd.

Calculation of eddy currents in moving structures using a finite element method on non‐matching grids

This paper deals with the numerical simulation of eddy current distributions in non‐stationary geometries with sliding interfaces. We study a system composed of two solid parts: a fixed one (stator) and a moving one (rotor) which slides in contact with the former. We also consider a two‐dimensional mathematical model based on the transverse electric formulation of the eddy current problem whose approximation is performed via the mortar element method combined with the standard linear finite element discretization in space and an implicit first order Euler scheme in time. Numerical results underline the influence of the rotor movement on the current distribution and give an estimate of the power losses with respect to the rotor angular speed.

A Quasi-Monte Carlo Scheme Using Nets for a Linear Boltzmann Equation

A quasi-Monte Carlo particle simulation for solving a linear Boltzmann equation is constructed and a convergence proof is given. The analysis is restricted to the three-dimensional equation in the space homogeneous case and the velocity domain is normalized to be I3 = [0,1)3. A particle simulation is described. It combines a Euler scheme in time with quasi-Monte Carlo integration in velocity space. The quadratures use (0,m,s)-nets, which are sets with a very regular distribution behavior. The particles are reordered according to the components of their velocity at each time step. A deterministic error bound is obtained. Finally, numerical experiments are presented for some test problems where an exact solution is known. The accuracy of the quasi-Monte Carlo scheme is compared with a standard Monte Carlo algorithm. The results show an improvement in both magnitude of error and convergence rate of quasi-Monte Carlo over Monte Carlo approach.

L ?-Error estimate for an approximation of a parabolic variational inequality

Almost optimalL∞-convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase Stefan problem. The discretization employs piecewise linear finite elements in space and the backward Euler scheme in time. By means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation. The latter bound is obtained by using the smoothing property of the Galerkin method.

Error estimates for two-phase stefan problems in several space variables, II: Non-linear flux conditions

In this paper we derive L2-error estimates for multidimensional two-phase Stefan problems involving further non-linearities such as non-linear flux conditions. We approximate the enthalpy formulation by a regularization procedure combined with a C0-piecewise linear finite element scheme in space, and the implicit Euler scheme in time.