A Posteriori Error Analysis for Implicit–Explicit hp-Discontinuous Galerkin Timestepping Methods for Semilinear Parabolic Problems

Abstract

A posteriori error estimates in the L ∞ (H)and L 2 (V)-norms are derived for fully-discrete space-time methods discretising semilinear parabolic problems; here V → H → V * denotes a Gelfand triple for an evolution partial differential equation problem. In particular, an implicit-explicit variable order (hp-version) discontinuous Galerkin timestepping scheme is employed, in conjunction with conforming finite element discretisation in space. The nonlinear reaction is treated explicitly, while the linear spatial operator is treated implicitly, allowing for time-marching without the need to solve a nonlinear system per timestep. The main tool in obtaining these error estimates is a recent space-time reconstruction proposed in Georgoulis et al. (A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems, Submitted for publication) for linear parabolic problems, which is now extended to semilinear problems via a non-standard continuation argument. Some numerical investigations are also included highlighting the optimality of the proposed a posteriori bounds. Keywords A posteriori error analysis • Semilinear PDEs • hp-discontinuous Galerkin timestepping • Space-time finite element methods • Reconstructions • Implicit-explicit timestepping methods B Emmanuil H.

A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems

“…Our main results include computable L ∞ (H )and L 2 (X )-a posteriori error estimates; some remarks concerning H 1 (X )-norm error estimation are given as well. Finally, we note that a series of numerical experiments showcasing the optimality of the a posteriori error estimators derived above are given in [7].…”

“…Elliptic reconstruction, nonetheless, offers the ability to use various elliptic estimators from the literature in the bound: this feature may become important for multiscale operators A , e.g., singular perturbations. In general, elliptic reconstruction allows for, crucial in some cases, flexibility in the handling for more complicated spatial operators A , e.g., nonlinear or singularly perturbed operators [6,7,18].…”

Conditional a posteriori error bounds for high order DG time stepping approximations of semilinear heat models with blow-up

“…By contrast, a posteriori error estimators for nonlinear problems are often conditional, that is, the error bounds only hold under the provision that some a posteriori verifiable condition is fulfilled. Most of the conditional estimates in the literature are explicit [6,10,11,19,21,26,30,31,35,39] in the sense that the estimates only hold under conditions of an explicit nature involving the magnitude of the numerical solution, the discretisation parameters and/or the problem data. Recently, there has been interest in the derivation of implicit conditional estimates, cf.…”