A discontinuous Galerkin method for higher-order ordinary differential equations

✍ Scribed by Slimane Adjerid; Helmi Temimi

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 369 KB
• Volume: 197
• Category: Article
• ISSN: 0045-7825
• DOI: 10.1016/j.cma.2007.07.015

✦ Synopsis

In this paper, we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal O(Dt p+1 ) convergence rate in the L 2 norm. We further show that the p-degree discontinuous solution of differential equation of order m and its first m À 1 derivatives are O(Dt 2p+2Àm ) superconvergent at the end of each step. We also establish that the p-degree discontinuous solution is O(Dt p+2 ) superconvergent at the roots of (p + 1 À m)-degree Jacobi polynomial on each step. Finally, we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates.