## 1 Introduction  
First-order Initial Value Problems (IVPs) play a fundamental role in theoretical and numerical analyses of physical and engineering problems [1; 2; 3; 4; 5]. Theoretically, any higher order ordinary differential equations (ODEs) can equivalently be transformed into systems of first-order ODEs and hence successful numerical methods for the first-order IVPs can also be applied to the Boundary Value Problems (BVPs) in ODEs. Numerically, it is often easier to develop straightforward, flexible and robust methods since only the solution functions are solved for without the need to consider solution derivatives. For adaptive analysis of the first-order IVPs, error estimation is required in certain kinds of norm. Instead of a thorough survey of various relevant numerical methods, the most used Runge-Kutta-Fehlberg method [3; 6] is worth mentioning on account of its representativeness and the availability of algorithms for adaptive analysis. The conventional adaptive approach based on the Runge-Kutta method estimates the truncation errors of a fourth-order Runge-Kutta method using the solution of a fifth-order Runge-Kutta method. This approach is simple and effective and can effectively control the truncation errors to a certain extent. However, the solution obtained by this method is in a discrete form, and it is impossible to measure the errors by maximum norm, and it is difficult to use the \(L^{2}\) norm as well. Moreover, this method is not \(A\)-stable for first-order IVPs [6], which flaws its flexibility and reliability. Although many research works on this method have been conducted to improve this method in recent years [7; 8; 9], the above-mentioned problems have not been fundamentally resolved. The conventional Galerkin finite element method (FEM) [10; 11; 12] is a convenient and effective tool for both BVPs and IVPs [4; 13; 14; 15; 16; 17; 18; 19]. Although, for the second-order equation of motion in structural dynamics analysis, the conventional Galerkin FEM can only generate conditionally stable algorithms, it can generate unconditionally stable algorithms if the second-order IVPs are transformed to the first-order IVPs in system of ODEs [13; 17; 20]. Specifically, characterized by piecewise polynomial spaces of degree \(m\), the conventional Galerkin FEM for the first-order IVPs [13; 17] can generate a one-step and \(A\)-stable method with \(O(h^{m+1})\) accuracy within elements, and \(O(h^{2m})\) accuracy at mesh points (element end nodes) [13] for nodal solutions (\(h\) being the maximum element size), which is particularly advantageous for a procedure of the time integration type. Recently, the authors of this paper proposed a condensed Galerkin element [20] for the first-order IVPs, which has been proved to be able to produce exactly identical nodal solutions as the conventional element of degree \(m+1\) for the IVPs with constant coefficients, and hence gains super-convergent nodal solutions of \(O(h^{2m+2})\); i.e., \(m\)-degreed elements gain the nodal results of \((m+1)\)-degreed elements. For adaptive finite element (FE) analysis, unlike discrete methods, the FE method provides continuous solutions on the complete time domain and hence it is possible to employ an error estimator in the maximum norm, which is exclusively adopted in this paper. A more significant aspect of the FEM is its super-convergent nodal solutions, i.e., \(O(h^{2m})\) for the conventional element or \(O(h^{2m+2})\) for the condensed element, because the nodal solutions serve as the initial values for the next time-step (element) solution. However, as is well known, element accuracy is not the only criterion for overall performance. Based on numerous theoretical analyses and numerical experiments conducted by the authors, it has been identified that, for a reliable and robust adaptive FE analysis of the first-order IVPs with adaptive time-stepping (element sizing) controlled by the maximum norm, there are three crucial requirements for a high-performance algorithm to satisfy, described as follows. **R1.