A reconstructed discontinuous approximation for distributed elliptic control problems

TL;DR

This paper introduces a high-order discontinuous Galerkin method with a reconstructed approximation for distributed elliptic control problems, providing error estimates and an efficient solution scheme verified by numerical experiments.

Contribution

It develops a novel reconstructed discontinuous Galerkin method with high-order approximation and analyzes its error estimates and computational efficiency.

Findings

The method achieves optimal convergence rates.

The a posteriori error estimates effectively guide adaptivity.

Numerical results confirm theoretical error bounds.

Abstract

In this paper, we present and analyze an interior penalty discontinuous Galerkin method for the distributed elliptic optimal control problems. It is based on a reconstructed discontinuous approximation which admits arbitrarily high-order approximation space with only one unknown per element. Applying this method, we develop a proper discretization scheme that approximates the state and adjoint variables in the approximation space. Our main contributions are twofold: (1) the derivation of both a priori and a posteriori error estimates of the $L^2$-norm and the energy norms, and (2) the implementation of an efficiently solvable discrete system, which is solved via a linearly convergent projected gradient descent method. Numerical experiments are provided to verify the convergence order in a priori error estimate and the efficiency of a posteriori error estimate.