A Discontinuous Galerkin Method for Higher-Order Ordinary Differential Equations Applied to the Wave Equation

(ABSTRACT) We propose a new discontinuous finite element method for higher-order initial value problems where the finite element solution exhibits an optimal convergence rate in the L 2 norm. We further show that the q-degree discontinuous solution of a differential equation of order m and its first m − 1 derivatives are strongly O(∆t 2q+2−m) superconvergent at the end of each step. We also establish that the q-degree discontinuous solution is O(∆t q+2) superconvergent at the roots of (q + 1 − m)-degree Jacobi polynomial on each step. Furthermore, we use these results to construct asymptotically correct a posteriori error estimates. Moreover, we design a new discontinuous Galerkin method to solve the wave equation by using a method of lines approach to separate the space and time where we first apply the classical finite element method using p-degree polynomials in space to obtain a system of second-order ordinary differential equations which is solved by our new discontinuous Galerkin method. We provide an error analysis for this new method to show that, on each space-time cell, the discontinuous Galerkin finite element solution is O(∆x p+2) + O(∆t q+2) superconvergent at the tensor product of the shifted roots of the Lobatto polynomials in space and the Jacobi polynomial P 2,0 q−1 (t) in time. Then, we show that the global L 2 error in space and time is O(∆x p+1) + O(∆t q+1) convergent. Furthermore, we are able to construct asymptotically correct a posteriori error estimates for both spatial and temporal components of errors. We validate our theory by presenting several computational results for one, two and three dimensions. Dedication I would like to dedicate this work to my father Bechir, my mother Hajer, my brother Yosri and my wife Zayneb. iii Acknowledgments I would like to declare that I owe my success and achievement to God, without whom I would not be able to reach this level of education. I am grateful to my advisor Dr. Slimane Adjerid for his scientific and moral support. For me Dr. Slimane was more than my advisor. Since our first days in Blacksburg, he provided us, my dear friend Mahboub and I, all the conditions to guarantee our success. More than that, he considered us as member of his family and with him We didn't feel like strangers away from our homes. Words cannot express my love and respect to Dr.Slimane. I would like to …