Irene Kyza

Time-discrete higher order ALE formulations: a priori error analysis

We derive optimal a priori error estimates for discontinuous Galerkin (dG) time discrete schemes of any order applied to an advection–diffusion model defined on moving domains and written in the Arbitrary Lagrangian Eulerian (ALE) framework. Our estimates hold without any restrictions on the time steps for dG with exact integration or Reynolds’ quadrature. They involve a mild restriction on the time steps for the practical Runge–Kutta–Radau methods of any order. The key ingredients are the stability results shown earlier in Bonito et al. (Time-discrete higher order ALE formulations: stability, 2013) along with a novel ALE projection. Numerical experiments illustrate and complement our theoretical results.

Analysis for Time Discrete Approximations of Blow-up Solutions of Semilinear Parabolic Equations

We prove a posteriori error estimates for time discrete approximations, for semilinear parabolic equations with solutions that might blow up in finite time. In particular we consider the backward Euler and the Crank-Nicolson methods. The main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments. The final estimates we derive are conditional and lead to error control near the blow up time.

Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems

This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically, a posteriori error bounds are derived in the

A posteriori error control and adaptivity for Crank---Nicolson finite element approximations for the linear Schrödinger equation

L^{\infty}(L^2)+L^2(H^1)

hp -Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

-type norm for a first order in time implicit-explicit interior penalty discontinuous Galerkin in space discretization of the problem, although the theory presented is directly applicable to the case of conforming finite element approximations in space. The choice of the discretization in time is made based on a careful analysis of adaptive time-stepping methods for ODEs that exhibit finite time blow-up. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including one which exhibits regional blow-up.

A dG Approach to Higher Order ALE Formulations in Time

We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schrödinger-type equations, in the

Error control for time-splitting spectral approximations of the semiclassical Schrödinger equation

L^\infty (L^2)

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

L∞(L2)-norm. For the discretization in time we use the Crank–Nicolson method, while for the space discretization we use finite element spaces that are allowed to change in time. The derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of Schrödinger equations. The final estimates are obtained using energy techniques and residual-type estimators. Various numerical experiments for the one-dimensional linear Schrödinger equation in the semiclassical regime, verify and complement our theoretical results. The numerical implementations are performed with both uniform partitions and adaptivity in time and space. For adaptivity, we further develop and analyze an existing time-space adaptive algorithm to the cases of Schrödinger equations. The adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant.