A Posteriori Error Computations in Finite Element Method for Initial Value Problems

Abstract

A posteriori error computations in the space-time coupled and space-time decoupled finite element methods for initial value problems are essential: 1) to determine the accuracy of the computed evolution, 2) if the errors in the coupled solutions are higher than an acceptable threshold, then a posteriori error computations provide measures for designing adaptive processes to improve the accuracy of the solution. How well the space-time approximation in each of the two methods satisfies the equations in the mathematical model over the space-time domain in the point wise sense is the absolute measure of the accuracy of the computed solution. When L 2 -norm of the space-time residual over the space-time domain of the computations approaches zero, the approximation ϕ h ( x , t ) → ϕ ( x , t ) , the theoretical solution. Thus, the proximity of ‖ E ‖ L 2 , the L 2 -norm of the space-time residual function, to zero is a measure of the accuracy or the error in the computed solution. In this paper, we present a methodology and a computational framework for computing ‖ E ‖ L 2 in the a posteriori error computations for both space-time coupled and space-time decoupled finite element methods. It is shown that the proposed a posteriori computations require h , p , k framework in both space-time coupled as well as space-time decoupled finite element methods to ensure that space-time integrals over space-time discretization are Riemann, hence the proposed a posteriori computations can not be performed in finite difference and finite volume methods of solving initial value problems. High-order global differentiability in time in the integration methods is essential in space-time decoupled method for posterior computations. This restricts the use of methods like Euler’s method, Runge-Kutta methods, etc., in the time integration of ODE’s in time. Mathematical and computational details including model problem studies are presented in the paper. To authors knowledge, it is the first presentation of the proposed a posteriori error computation methodology and computational infrastructure for initial value problems.