The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part II: A Posteriori Error Estimation

Abstract

In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler–Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to $(k+1)$-degree Radau polynomials, when polynomials of total degree not exceeding $k$ are used. These results allow us to prove that the $k$-degree LDG solution and its derivatives are $\mathcal O (h^{k+3/2})$ superconvergent at the roots of $(k+1)$-degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time $t$ converge to the true errors at $\mathcal O (h^{k+5/4})$ rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the $L^2$-norm converge to unity at $\mathcal O (h^{1/2})$ rate. Our proofs are valid for arbitrary regular meshes and for $P^k$ polynomials with $k\ge 1$, and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.