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

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 ${？}_h\left(x,t\right)\to ？\left(x,t\right)$ , the theoretical solution. Thus, the proximity of ${\Vert E\Vert }_{L_2}$ , 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 ${\Vert E\Vert }_{L_2}$ 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