Two Efficient and Reliable a posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Linear Elliptic Problems on Cartesian Grids

In this paper, we derive two a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to linear second-order elliptic problems on Cartesian grids. We first prove that the gradient of the LDG solution is superconvergent with order \(p+1\) towards the gradient of Gauss-Radau projection of the exact solution, when tensor product polynomials of degree at most p are used. Then, we prove that the gradient of the actual error can be split into two parts. The components of the significant part can be given in terms of \((p+1)\)-degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We further develop a postprocessing gradient recovery scheme for the LDG solution. This recovered gradient superconverges to the gradient of the true solution. The order of convergence is proved to be \(p+1\). We use our gradient recovery result to develop a robust recovery-type a posteriori error estimator for the gradient approximation which is based on an enhanced recovery technique. We prove that the proposed residual-type and recovery-type a posteriori error estimates converge to the true errors in the \(L^2\)-norm under mesh refinement. The order of convergence is proved to be \(p + 1\). Moreover, the proposed estimators are proved to be asymptotically exact. Finally, we present a local adaptive mesh refinement procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for \(P^p\) polynomials with \(p\ge 1\). We provide several numerical examples illustrating the effectiveness of our procedures.