*3.4. Computational Complexity Analysis*

Finally, a computational complexity of the non-reduced and reduced DPL models was investigated. First, an analysis of the time consumed during the solution calculation was carried out and is shown in Figure 8. It can be seen that the smaller the distance between mesh nodes the longer the time needed to obtain a temperature distribution in the investigated cross-sectional area of the MEMS structure. This is caused by a greater number of nodes in the smaller considered node distances.

**Figure 8.** Comparison of simulation time consumed during solution calculation in relation to discretization mesh nodes distance based on non-reduced and reduced DPL models.

The simulation time investigation in relation to the number of nodes in the analyzed cross-section is shown in Figure 9. The significant difference in simulation time needed for results preparation is visible. Considering the non-reduced DPL model, on the one hand, the simulation time can be estimated by a power function according to Equation (23). On the other hand, the time consumed using the reduced DPL model can be approximated based on a linear function presented in Equation (24). Moreover, statistics describing Equations for simulation time estimations are shown in Table 2.

$$t\_{non-reduced}^{DPL}(n) = 1.96 \cdot 10^{-7} \cdot n^{2.544} \tag{23}$$

$$t\_{reduced}^{DPL}(n) = 2.546 \cdot 10^{-5} \cdot n - 0.03782\tag{24}$$

where n reflects the number of nodes in the considered MEMS structure cross-section.

**Figure 9.** Comparison of the simulation time consumed during the solution calculation in relation to the number of nodes in the analyzed cross-section of the MEMS structure based on the non-reduced and reduced DPL models.


**Table 2.** Simulation time approximation.

The values of metrics presented in Table 2 confirm the high quality of the prepared Equations for simulation time estimation. Moreover, based on Equations (23) and (24) and the simulation time analysis, it can be stated that the time complexity of an algorithm using the non-reduced DPL model is O(*n2.544*), whereas the reduced DPL is characterized by a time complexity O(*n*). It is also worthwhile highlighting that the DPL system Equation (21) is linear (regarding the time variable), and therefore the Krylov subspace is calculated only once. Thus, it is not updated in each time step. Of course, the computational complexity of the method includes considerations regarding the Krylov subspace generation.
