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 L ∞ (L 2 ) + L 2 (H 1 )-type norm for a first order in time implicit-explicit (IMEX) interior penalty discontinuous Galerkin (dG) 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.

Abstract. Arbitrary Lagrangian Eulerian (ALE) formulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. This arbitrary extension has no effect on the stability of the PDE but may influence that of a discrete scheme. We propose time-discrete discontinuous Galerkin (dG) numerical schemes of any order for a time-dependent advection-diffusion model problem in moving domains, and study their stability properties. The analysis hinges on the validity of the Reynolds' identity for dG. Exploiting the variational structure and assuming exact integration, we prove that our conservative and non-conservative dG schemes are equivalent and unconditionally stable. The same results remain true for piecewise polynomial ALE maps of any degree and suitable quadrature that guarantees the validity of the Reynolds' identity. This approach generalizes the so-called geometric conservation law (GCL) to higher order methods. We also prove that simpler Runge-Kutta-Radau (RKR) methods of any order are conditionally stable, that is subject to a mild ALE constraint on the time steps. Numerical experiments corroborate and complement our theoretical results.

Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations Irene Kyza and Charalambos MakridakisOriginal Citation:Kyza, Irene and Makridakis, Charalambos http://preprints.acmac.uoc.gr/ ANALYSIS FOR TIME DISCRETE APPROXIMATIONS OF BLOW-UP SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS IRENE KYZA AND CHARALAMBOS MAKRIDAKISAbstract. 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.

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated.

Abstract. 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 (RKR) methods of any order. The key ingredients are the stability results shown earlier in [6] along with a novel ALE projection. Numerical experiments illustrate and complement our theoretical results.