Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD–Greedy...

We consider the Cauchy problem in an unbounded region for equations of the type either $D_{t}z(t,x)=f(t,x,z(t,x),z_{(t,x)},D_{x}z(t,x))$ or $D_{t}z(t,x)=f(t,x,z(t,x),z,D_{x}z(t,x))$. We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.

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...

We study convergence of solutions to stationary states in an astrophysical model of evolution of clouds of self-gravitating particles.

We study the behaviour of the solutions of the Cauchy problem $$u_t={\left(u^m\right)}_{xx}+u\left(1-u^{m-1}\right),x\in \mathbb{R},t>0u\left(0,x\right)=u_0\left(x\right),u_0\left(x\right)\ge 0,$$ and prove that if initial data $u_0\left(x\right)$ decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.

This work proves the convergence in L¹(ℝ²) towards an Oseen vortex-like solution to the dissipative quasi-geostrophic equations for several sets of initial data with suitable decay at infinity. The relative entropy method applies in a direct way for solving this question in the case of signed initial data and the difficulty lies in showing the existence of unique global solutions for the class of initial data for which all properties needed in the entropy approach are met. However, the estimates...

In this paper, two algorithms are proposed to solve systems of algebraic equations generated by a discretization procedure of the weak formulation of boundary value problems for systems of nonlinear elliptic equations. The first algorithm, Newton-CG-MG, is suitable for systems with gradient mappings, while the second, Newton-CE-MG, can be applied to more general systems. Convergence theorems are proved and application to the semiconductor device modelling is described.

We propose and analyse two convergent fully discrete schemes to solve the incompressible Navier-Stokes-Nernst-Planck-Poisson system. The first scheme converges to weak solutions satisfying an energy and an entropy dissipation law. The second scheme uses Chorin's projection method to obtain an efficient approximation that converges to strong solutions at optimal rates.

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 paper we present a few results on convergence for the prime integrals equations connected with the bounce problem. This approach allows both to prove uniqueness for the one-dimensional bounce problem for almost all permissible Cauchy data (see also [6]) and to deepen previous results (see [3], [5], [7]).

This paper reviews popular acceleration techniques to converge the non-linear self-consistent field equations appearing in quantum chemistry calculations with localized basis sets. The different methodologies, as well as their advantages and limitations are discussed within the same framework. Several illustrative examples of calculations are presented. This paper attempts to describe recent achievements and remaining challenges in this field.