Superconvergence of the local discontinuous galerkin method applied to the one‐dimensional second‐order wave equation. Issue 3 (23rd December 2013) - Baccouch, Mahboub
<x xml:space="preserve">Abstract</x> We analyze the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to the second‐order wave equation in one space dimension. With a suitable projection of the initial conditions for the LDG scheme, we prove that the LDG solution and its spatial derivative are <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kp37" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />O(hp+3/2)</alternatives> super close to particular projections of the exact solutions for pth‐degree polynomial spaces. We use these results to show that the significant parts of the discretization errors for the LDG solution and its derivative are proportional to <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kp14" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />(p+1)</alternatives>‐degree right and left Radau polynomials, respectively. These results allow us to prove that the p‐degree LDG solution and its derivative are <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kq03" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />O(hp+3/2)</alternatives> superconvergent at the roots of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kq1n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />(p+1)</alternatives>‐degree right and left Radau polynomials, respectively, while computational results show higher <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kpxg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />O(hp+2)</alternatives> convergence rate. Superconvergence results can be used to construct asymptotically correct a posteriori error estimates by solving a local steady problem on each element. This will be discussed further in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivative converge to the true spatial errors in the L2‐norm under mesh refinement. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 862–901, 2014