On multiresolution methods in numerical analysis.
On some composite schemes of time integration in structural dynamics
Numerical simulations of time-dependent behaviour of advances structures need the analysis of systems of partial differential equations of hyperbolic type, whose semi-discretization, using the Fourier multiplicative decomposition together with the finite element or similar techniques, leads to large sparse systems of ordinary differential equations. Effective and robust methods for numerical evaluation of their solutions in particular time steps are required; thus still new computational schemes...
On the discretization of a partial differential equation in the neighborhood of a periodic orbit.
On the method of lines for a non-linear heat equation with functional dependence
We consider a heat equation with a non-linear right-hand side which depends on certain Volterra-type functionals. We study the problem of existence and convergence for the method of lines by means of semi-discrete inverse formulae.
On the numerical solution of the diffusion equation with a nonlocal boundary condition.
Optimal error estimates for semidiscrete phase relaxation models
Optimal grids for anisotropic problems.
Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies
In this paper, we investigate the coupling between operator splitting techniques and a time parallelization scheme, the parareal algorithm, as a numerical strategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive...
Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies
In this paper, we investigate the coupling between operator splitting techniques and a time parallelization scheme, the parareal algorithm, as a numerical strategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive...
Resolvent estimates in for discrete Laplacians on irregular meshes and maximum-norm stability of parabolic finite difference schemes.
Resolvent estimates of elliptic differential and finite-element operators in pairs of function spaces.
Rothe time-discretization method for the semilinear heat equation subject to a nonlocal boundary condition.
Rothe's method for degenerate quasilinear parabolic equations
Runge-Kutta methods for parabolic equations with nonsmooth initial data
Semidiscrete and single step fully discrete finite element approximations for second order hyperbolic equations with nonsmooth solutions
Semidiscretization for a nonlocal parabolic problem.
Semigroup formulation of Rothe's method: application to parabolic problems
A semilinear parabolic equation in a Banach space is considered. The purpose of this paper is to show the dependence of an error estimate for Rothe's method on the regularity of initial data. The proofs are done using a semigroup theory and Taylor spectral representation.
Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations , in ; fully coupled by the boundary conditions , on , where is a bounded smooth domain in . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation . We prove that if blows up in finite time then can fail to blow up if and only if and , which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover,...
Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (0,T); fully coupled by the boundary conditions , on ∂Ω x (0,T), where Ω is a bounded smooth domain in . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U,V). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1) , which is the same condition as...
Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data
The purpose of this paper is to derive the error estimates for discretization in time of a semilinear parabolic equation in a Banach space. The estimates are given in the norm of the space for when the initial condition is not regular.