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

We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method dG(q) and the corresponding implicit RungeKutta-Radau method IRK-R(q) of arbitrary order q ≥ 0 for both linear and nonlinear evolution problems of the form u′ + F(u) = f , with γ-angle bounded operator F. The key ingredient is a novel higher order reconstruction Û of the discrete solution U , which restores continuity and leads to the differential equation Û ′ + ΠF(U) = F for a suitable interpolation operator Π and piecewise polynomial approximation F of f . We discuss applications to linear PDE, such as the convection-diffusion equation (γ ≥ 12 ) and the wave equation (formally γ = ∞), and nonlinear PDE corresponding to subgradient operators (γ = 1), such as the p-Laplacian, as well as Lipschitz operators (γ ≥ 12 ). We also derive conditional a posteriori error estimates for the time-dependent minimal surface problem.