An adaptive finite element method for systems of second-order hyperbolic partial differential equations in one space dimension.

In this dissertation I study an adaptive finite element method for vector systems of second-order hyperbolic partial differential equations in one space dimension. To reduce the computational cost, explicit methods are generally preferred for hyperbolic problems. As a result the finite element method of lumped masses instead of consistent masses has been implemented to discretize the wave equation. Lagrange basis functions and Gauss-Lobatto quadrature rule are used to obtain mass lumping. The diagonal mass matrix resulting from lumping leads to a special system of second-order ordinary differential equations. Instead of transforming this system to an equivalent first-order system as is typically done, I solve this directly by using explicit Runge-Kutta-Nystrom method that offers improved effciency and less memory. The goal of this work is to determine an error estimate that best indicates the refinement regions and/or selects the order of the basis functions in the adaptive procedure. Two explicit and one implicit interpolation-error based a posteriori error estimates developed for elliptic and parabolic problems have been extended to hyperbolic problems and used to estimate the spatial discretization error of the wave equation. One of the explicit estimators involves high-order derivatives of the computed solution and the other uses the first (second) derivative jumps of the computed solution at the element boundaries for odd (even) order bases. The implicit estimator uses auxiliary equations. All of these estimators are capable of computing estimates at several orders including one order higher than the current order, and thus, are appropriate for an order variation strategy. They are also seen to be asymptotically exact for the method of lumped masses for second-order hyperbolic problems. These estimators drive the hp-adaptive strategy. Computational results demonstrate the effectiveness of all estimators and make possible a comparison of them.