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

Abstract

We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method dG(q) and the corresponding implicit Runge- Kutta-Radau method IRK-R(q) of arbitrary order q≥0 for both linear and nonlinear evolution problems of the form $$u^{\prime} + \mathfrak{F}(u) = f$$, with γ2-angle bounded operator $$\mathfrak{F}$$. The key ingredient is a novel higher order reconstruction $$\widehat{U}$$ of the discrete solution U, which restores continuity and leads to the differential equation $$\widehat{U}^{\prime}+\Pi\mathfrak{F}(U)=F$$ 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 ≥ 1/2) 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 ≥ 1/2). We also derive conditional a posteriori error estimates for the time-dependent minimal surface problem.