Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems

Abstract

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.

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

“…Thus the methodology of this section cannot be generalised to the focusing NLS equation with the aim to control the error close to the blowup time. In contrast to the corresponding parabolic equation with blowup, cf., [15], energy methods are not appropriate for the focusing NLS with blowup and other techniques should be applied for the a posteriori error analysis of these equations.…”

“…Estimate (3.17) may be used for the proposition of an efficient adaptive algorithm as in [15,34]. This estimate is more appropriate for adaptivity due its local nature.…”

“…If this term is large, again the estimates will be unpractical for uniform partitions. However, as in the case of [15,34], the proposition of an efficient time-space adaptive algorithm (using the local estimate (3.20)) could provide a way to control the error.…”

“…A similar term appears in the a posteriori error analysis for semilinear parabolic equations with possible blowup in finite time; cf. [15,32]. In fact, as illustrated in [15,34], this term enables the proposition of an efficient time-space adaptive algorithm leading to blowup detection and accurate numerical approximation of blowup times.…”

“…In fact, as illustrated in [15,34], this term enables the proposition of an efficient time-space adaptive algorithm leading to blowup detection and accurate numerical approximation of blowup times. In the cases of NLS equations (1.1) we expect that the exponential term will also be proven beneficial towards the development of an efficient time-space adaptive algorithm in the spirit of [15]. This is the subject of a forthcoming paper.…”