Super-convergence and Stability Analysis of the Finite Element Orthogonal Collocation Method for Time-Fractional Telegraph Equation

Super-convergence and stability analysis are essential components for ensuring efficiency, reliability, and accuracy of numerical approximations to iterative methods and differential equations. Stability analysis highlights the mechanism of error control over iterations to ensure reliability and long-term accuracy of simulation. Super-convergence analysis offers some extra conditions to ensure faster convergence of numerical solutions than expected, enabling enhanced accuracy and strategies. These analyses together guarantee the effective development of complex numerical algorithms as the basis for error control and estimation. These analyses involve complex simulations and are thus relevant in fields such as physics, engineering, and weather modeling. Thus, this paper is an extension of the Finite Element Orthogonal Collocation Method (FEOCM) by Mamadu et al. (2023) for the numerical approximation of the time fractional telegraph equation with Mamadu-Njoseh polynomials as basis functions, where relevant numerical simulations were carried out. The present study offers a comprehensive super-convergence and stability analysis of solutions for the time-fractional telegraph equations via FEOCM. Here, the L₂-norm, H¹-norm, interpolation theory, and Cauchy-Schwarz inequality are employed as optimal estimators to propose relevant theorems for the analysis of stability and super-convergence of solutions. The analysis shows that the solutions of the fully discretized scheme FEOCM are unconditionally stable and exhibit super-convergence, with the optimal error estimated as <img src=image/13438864_01.gif>.