This work proposes a finite element scheme with P1 and P2 basis functions using an Euler-Taylor-Galerkin method for a nonlinear model that describes the behavior of a new chemo-fluidic oscillator. This model is expressed through the coupling of an ordinary differential equation describing the hydrogel dynamics, a nonlinear transport equation, and an auxiliary equation determining the flow volume. The numerical solution is constructed by taking a semi-discretization in time of the transport equation using forward Taylor series expansions that include second and third-order time derivatives to avoid instability issues. In this semi-discrete equation, the spatial variable is approximated using a Galerkin finite element formulation. Several simulations are performed with different initial conditions for the hydrogel concentration. Additionally, the performance of the Euler-Taylor-Galerkin method is compared with other popular numerical methods, such as the Finite Volume Method and the Finite Difference Method, in terms of stability, accuracy, and computational efficiency. The results of this comparison demonstrate the advantages of the Euler-Taylor-Galerkin method in terms of stability and accuracy for problems with complex geometries and significant nonlinearities. Furthermore, a sensitivity analysis is conducted to evaluate how changes in system parameters (such as initial alcohol concentration, temperature, etc.) affect the oscillator’s behavior, helping to understand the system’s robustness against external perturbations. This analysis shows that the chemo-fluidic oscillator is robust against moderate variations in system parameters, although significant perturbations can drastically affect its behavior. The numerical results describe the oscillatory behavior of the system, and Matlab tools are used for simulations, providing robust validation of the proposed model.