A discontinuous Galerkin method for higher-order ordinary differential equations (Q839179)

This paper is concerned with the application of the Discontinuous Galerkin (DG) method introduced by \textit{P. Lesaint} and \textit{P. Raviart} [On a finite element method for solving the neutron transport equations. in: Mathematical aspects of finite elements in partial differential equations, Academic Press, New York, 89--145 (1974)], in the context of PDEs to the numerical solution of mth order \((m > 1)\) initial value problems for ordinary differential equations. Although any mth order equation can be transformed to into a first order system where the solution and its first \((m-1)\) derivatives are treated as dependent variables and this system can be solved by DG methods, here the authors only solve for the solution of the mth order IVP. It is proved that the piecewise \(p\)--degree DG finite element solution with \( p > m \) is \( O ( \Delta t^{p+1} )\) to the true solution in the two--norm. In addition they show that such a solution is superconvergent at the roots of the \((p+1-m)\)--degree Jacobi polynomial. Finally, the solution and its first \(m-1\) derivatives are \( O ( \Delta t^{2p+2-m} )\) superconvergent at the end of an step and asymptotically correct expression of the leading error term is also derived.