## 6 Numerical Examples
### A Dynamic Problem  
The problem to be solved is a typical single degree-of-freedom problem [22] posed in a second-order ODE form as follows:  
\[u^{\prime\prime}+0.04u^{\prime}+u=\sin(0.2t),\ u(0)=0,\ u^{\prime}(0)=1,\ 0<t \leq 256\text{s}, \tag{38}\]  
which is typically an equation of motion and physically represents a damped harmonic motion subjected to a dynamic force. The exact solution is shown in Figure 15. By introducing \(u_{1}=u\) and \(u_{2}=u^{\prime}\), the problem can be equivalently transformed to a system of first-order ODEs:  
\[u_{1}^{\prime}=u_{2},\ u_{2}^{\prime}+0.04u_{2}+u_{1}=\sin(0.2t),\ u_{1}(0)=0, \ u_{2}(0)=1,\ 0<t\leq 256\text{s}, \tag{39}\]  
which is solved by using the proposed algorithm with the reduced elements. Please note that, to test the stability of the algorithm for long time domains, a time domain of 256 s has been taken deliberately. The computed results are shown in Table 8 and the adapted step sizes and error distributions are shown in Figures 16 and 17, respectively. It is seen that all the FE solutions strictly satisfy the tolerances. It is also observed that the element size has changed at most five times and the number of adaptive iterations is no more than six times.  
To access the performance of the reduced element compared with the standard (conventional) elements, the problem has also been solved by using the standard linear and quadratic Galerkin elements (see Section 3) with the same adaptive step-size algorithm using the EEP (Element Energy Projection) [16; 17; 18; 19] super-convergent solution as the error estimator. The computed results are shown in Table 9 and the adapted step sizes and error distributions are shown in Figure 18. It is seen that both standard elements fail to control the error to meet the given tolerance, and that the maximum error ratios are as high as 15.9 for the linear element and 5.63 for the quadratic element. The key reason is that neither of the accuracy ratios reaches two (one for the linear element and four thirds for the quadratic element), and hence neither of them can effectively provide satisfactory results.  
Figure 15: The exact solution of the dynamic problem.  
Figure 16: The step-size distribution of the dynamic problem (\(m=3\), \(T_{l}=10^{-3}\)).  
Figure 17: The error ratio variation of the dynamic problem (\(m=3\), \(T_{l}=10^{-3}\)).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Tolerance**} & \multirow{2}{*}{\begin{tabular}{c} **Num. of** \\ **Elements Used** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Num. of** \\ **Elements Used** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Num. of** \\ **Elements Used** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Num. of** \\ **Elements Sizes** \\ **Altered** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Num. of** \\ **Elements Sizes** \\ **Iterations** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Maximum** \\ **Element Size** \\ **Iterations** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Minimum** \\ **Element Size** \\ **Iterations** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **True Maximum** \\ **Element Error** \\ **Ratio** \\ \(-h\) \\ \(\epsilon\) **max** \\ \end{tabular} } \\ \hline \multicolumn{7}{c}{\(m=1\)} \\ \hline \(10^{-3}\) & 1542 & 41 & 46 & 0.7835 & 0.0087 & 0.90 \\ \(10^{-4}\) & 4831 & 88 & 91 & 0.3258 & 0.0290 & 0.85 \\ \(10^{-5}\) & 15413 & 122 & 125 & 0.0876 & 0.0090 & 0.85 \\ \(10^{-6}\) & 48539 & 146 & 149 & 0.0268 & 0.0029 & 0.85 \\ \hline \hline \multicolumn{7}{c}{\(m=3\)} \\ \hline \(10^{-3}\) & 162 & 4 & 4 & 2.69 & 0.87 & 0.78 \\ \(10^{-4}\) & 289 & 5 & 5 & 2.03 & 0.50 & 0.84 \\ \(10^{-5}\) & 498 & 5 & 6 & 0.92 & 0.33 & 0.83 \\ \(10^{-6}\) & 887 & 5 & 6 & 0.52 & 0.20 & 0.84 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Results of the dynamic problem (256 s).