We study space–time finite element methods for semilinear parabolic problems
in (1 + d)–dimensions for d = 2, 3. The discretisation in time is based on the
discontinuous Galerkin timestepping method with implicit treatment of the linear
terms and either implicit or explicit multistep discretisation of the zeroth order
nonlinear reaction terms. Conforming finite element methods are used for the
space discretisation. For this implicit-explicit IMEX–dG family of methods, we
derive a posteriori and a priori energy-type error bounds and we perform extended
numerical experiments. We derive a novel hp–version a posteriori error bounds in
the L1(L2) and L2(H1) norms assuming an only locally Lipschitz growth condition
for the nonlinear reactions and no monotonicity of the nonlinear terms. The
analysis builds upon the recent work in [60], for the respective linear problem,
which is in turn based on combining the elliptic and dG reconstructions in [83, 84]
and continuation argument. The a posteriori error bounds appear to be of optimal
order and efficient in a series of numerical experiments.
Secondly, we prove a novel hp–version a priori error bounds for the fully–discrete
IMEX–dG timestepping schemes in the same setting in L1(L2) and L2(H1) norms.
These error bounds are explicit with respect to both the temporal and spatial
meshsizes kn and h, respectively, and, where possible, with respect to the possibly
varying temporal polynomial degree r. The a priori error estimates are derived
using the elliptic projection technique with an inf-sup argument in time. Standard
tools such as Grönwall inequality and discrete stability estimates for fully discrete
semilinear parabolic problems with merely locally-Lipschitz continuous nonlinear
reaction terms are used. The a priori analysis extends the applicability of the
results from [52] to this setting with low regularity. The results are tested by an
extensive set of numerical experiments.
محمد السبعاوي | Mohammad Sabawi | 1514