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

Abstract

AbstractA posteriori error estimates in the $$L_\infty (\mathcal {H})$$L∞(H)- and $$L_2(\mathcal {V})$$L2(V)-norms are derived for fully-discrete space–time methods discretising semilinear parabolic problems; here $$ \mathcal {V}\hookrightarrow \mathcal {H}\hookrightarrow \mathcal {V}^*$$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.