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In this paper we present two versions of the central local
discontinuous Galerkin (LDG) method on overlapping cells
for solving diffusion equations, and provide their
stability analysis and error estimates for the linear heat equation.
A comparison
between the traditional LDG method on
a single mesh and the two versions of the central LDG
method on overlapping cells is also made.
Numerical experiments are provided to validate the quantitative
conclusions from the analysis and to support conclusions...
In this paper we present two versions of the central local
discontinuous Galerkin (LDG) method on overlapping cells
for solving diffusion equations, and provide their
stability analysis and error estimates for the linear heat equation.
A comparison
between the traditional LDG method on
a single mesh and the two versions of the central LDG
method on overlapping cells is also made.
Numerical experiments are provided to validate the quantitative
conclusions from the analysis and to support conclusions...
We present a reduced basis offline/online procedure for viscous Burgers initial boundary value problem, enabling efficient approximate computation of the solutions of this equation for parametrized viscosity and initial and boundary value data. This procedure comes with a fast-evaluated rigorous error bound certifying the approximation procedure. Our numerical experiments show significant computational savings, as well as efficiency of the error bound.
In this article, we present a numerical scheme based on a finite element method in order to solve a time-dependent convection-diffusion equation problem and satisfy some conservation properties. In particular, our scheme is able to conserve the total energy for a heat equation or the total mass of a solute in a fluid for a concentration equation, even if the approximation of the velocity field is not completely divergence-free. We establish a priori errror estimates for this scheme and we give some...
In this paper we analyze the consistency, the accuracy and some entropy properties of particle methods with remeshing in the case of a scalar one-dimensional conservation law. As in [G.-H. Cottet and L. Weynans, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 51–56] we re-write particle methods with remeshing in the finite-difference formalism. This allows us to prove the consistency of these methods, and accuracy properties related to the accuracy of interpolation kernels. Cottet and Magni devised recently...
In this paper we analyze the consistency, the accuracy and some entropy properties of
particle methods with remeshing in the case of a scalar one-dimensional conservation law.
As in [G.-H. Cottet and L. Weynans, C. R. Acad. Sci. Paris, Ser. I
343 (2006) 51–56] we re-write particle methods with remeshing in
the finite-difference formalism. This allows us to prove the consistency of these methods,
and accuracy properties related to the accuracy of...
We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.
In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.
We formulate a finite element method for the computation of solutions to an anisotropic phase-field model for a binary alloy. Convergence is proved in the -norm. The convergence result holds for anisotropy below a certain threshold value. We present some numerical experiments verifying the theoretical results. For anisotropy below the threshold value we observe optimal order convergence, whereas in the case where the anisotropy is strong the numerical solution to the phase-field equation does not...
The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly...
The problem of modeling acoustic waves scattered by an object with
Neumann boundary condition is considered. The boundary condition is
taken into account by means of the fictitious domain method, yielding
a first order in time mixed variational formulation for the
problem. The resulting system is discretized
with two families of mixed finite elements that are compatible with
mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always...
A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.
A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.
In this article, a method of cubic spline curve fitting to a set of points passing at a prescribed distance from input points obtained by measurement on a coordinate measuring machine is described. When reconstructing the shape of measured object from the points obtained by real measurements, it is always necessary to consider measurement uncertainty (tenths to tens of micrometres). This uncertainty is not zero, therefore interpolation methods, where the resulting curve passes through the given...
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