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

Abstract

Abstract. We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method and the corresponding implicit Runge-KuttaRadau method of arbitrary order for both linear and nonlinear evolution problems. The key ingredient is a novel higher order reconstruction U of the discrete solution U , which restores continuity and leads to the differential equation U +ΠF(U ) = F for a suitable interpolation operator Π. 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.

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

“…It shows that the error indicator η cG gives rise to a reliable and efficient a posteriori error estimate, up to data approximation terms. (19). Then the error measure (17) satisfies the upper bound…”

“…Applying the ideas in [19], and proceeding similarly to the analysis of the cG method, the following hp-version a posteriori error estimates are obtained.…”

“…For example, a posteriori error estimation based on duality techniques can be found in, e.g., [3,10,16] and the references therein. An alternative approach, applied to dG methods, has been recently presented in [19]. Here, the analysis is based on an appropriate reconstruction of the dG solution which allows the dG variational formulation to be rewritten in strong form, and subsequently, enables the application of natural energy arguments.…”

“…Here, the analysis is based on an appropriate reconstruction of the dG solution which allows the dG variational formulation to be rewritten in strong form, and subsequently, enables the application of natural energy arguments. In this paper, we shall use the h-version approach presented in [19], and extend it to the hp-version cG and dG methods. Specifically, we present a posteriori error estimators for both the hp-version cG and dG schemes which are fully explicit with respect to the size of the time-steps and polynomial degrees.…”

“…Specifically, we present a posteriori error estimators for both the hp-version cG and dG schemes which are fully explicit with respect to the size of the time-steps and polynomial degrees. The main novelties of the present analysis are: (1) a complete hp-characterization of the reconstruction operator from [19] and its difference (measured in a suitable norm) to the dG solution, and (2) the application of a similar approach to the hp-version cG scheme. The resulting error estimators are tested within an hp-adaptive algorithm based on local Sobolev regularity estimation, as proposed in [12] for elliptic problems.…”