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


Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA



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.