Title page for ETD etd-03182008-143719


Type of Document Dissertation
Author Temimi, Helmi
Author's Email Address temimi@vt.edu
URN etd-03182008-143719
Title A Discontinuous Galerkin Method for Higher-Order Differential Equations Applied to the Wave Equation.
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Adjerid, Slimane Committee Chair
Hagedorn, George A. Committee Member
Lin, Tao Committee Member
Sun, Shu-Ming Committee Member
Keywords
  • Superconvergence
  • Discontinuous Galerkin Method
  • a posteriori error estimation
  • wave equation
Date of Defense 2008-03-17
Availability unrestricted
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 L2- 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 superconvergent at the end of each step. We also establish that the q-degree discontinuous solution is 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 superconvergent at the tensor product of the shifted roots of the Lobatto polynomials in space and the Jacobi polynomial in time. Then, we show that the global L2 error in space and time is 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.

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