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

Kyza, Irene; Makridakis, Charalambos

doi:10.1137/100796819

Cited by 16 publications

(30 citation statements)

References 34 publications

(34 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[5,14], and fixed point arguments, cf. [6,15]. The a posteriori error bounds that we derive here are obtained by utilizing a local continuation argument.…”

Section: U(t) H < ∞ For 0 < T < T ∞ Lim T T ∞ U(t) H = ∞mentioning

confidence: 99%

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

2017

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…[5,14], and fixed point arguments, cf. [6,15]. The a posteriori error bounds that we derive here are obtained by utilizing a local continuation argument.…”

Section: U(t) H < ∞ For 0 < T < T ∞ Lim T T ∞ U(t) H = ∞mentioning

confidence: 99%

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

2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…A posteriori error estimators for finite element discretizations of nonlinear parabolic problems are available in the literature (e.g., [5,6,7,11,14,24,31,46,47,48]). However, the literature on a posteriori error control for parabolic equations that exhibit finite time blow-up is very limited; to the best of our knowledge, only in [33] do the authors provide rigorous a posteriori error bounds for such problems. Using a semigroup approach, the authors of [33] arrive to conditional a posteriori error estimates in the L ∞ (L ∞ )-norm for first and second order temporal semi-discretizations of a semilinear parabolic equation with polynomial nonlinearity.…”

mentioning

confidence: 99%

Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems

Cangiani

Georgoulis

Kyza³

et al. 2016

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…These estimates are based on subtle PDE arguments, see e.g. [23] and in particular [19] treating the case of a nonlinear parabolic equation with possible finite time blow-up, and are beyond the scope of the present paper.…”

Section: A Posteriori Estimates For the Semidiscrete Approximationmentioning

confidence: 99%