#
A posteriori error analysis for higher order dissipative methods for
evolution problems

Charalambos Makridakis

Department of Applied Mathematics

University of Crete

71409 Heraklion-Crete, Greece

and Institute of Applied and
Computational Mathematics

FORTH, 71110 Heraklion-Crete, Greece

makr@math.uoc.gr
Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
##
Abstract

We prove a posteriori error estimates for time discretizations by the
discontinuous Galerkin method and the corresponding implicit Runge-Kutta-Radau
method of arbitrary order for both linear and nonlinear evolution
problems. The key ingredient is a novel higher order reconstruction $\wU$
of the discrete solution $U$, which restores continuity and leads to
the differential equation $\wU'+\Pi\F(U)=F$ for a suitable interpolation
operator $\Pi$. The error analysis hinges
on careful energy arguments and the monotonicity of
the operator $\F$, in particular its
angle bounded structure. We discuss applications to
linear PDE such as the convection-diffusion equation
and the wave equation, and nonlinear PDE corresponding to subgradient
operators such as the $p$-Laplacian and minimal surfaces, as well as
Lipschitz and noncoercive operators.