TITLE:
The Coercive Property and a Priori Error Estimation of the Finite Element Method for Linearly Distributed Time Order Fractional Telegraph Equation with Restricted Initial Conditions
AUTHORS:
Ebimene James Mamadu, Henrietta Ify Ojarikre, Daniel Chinedu Iweobodo, Ebikonbo-Owei Anthony Mamadu, Jonathan Tsetimi, Ignatius Nkonyeasua Njoseh
KEYWORDS:
Coercivity, Finite Element Method, Mamadu-Njoseh Polynomials, A Priori Error Estimation, Cauchy-Schwarz Inequality, Mean Value Theorem
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.14 No.4,
October
15,
2024
ABSTRACT: Finite Element Method (FEM), when applied to solve problems, has faced some challenges over the years, such as time consumption and the complexity of assumptions. In particular, the making of assumptions has had a significant influence on the accuracy of the method, making it mandatory to carry out sensitivity analysis. The sensitivity analysis helps to identify the level of impact the assumptions have on the method. However, sensitivity analysis via FEM can be very challenging. A priori error estimation, an integral part of FEM, is a basic mathematical tool for predicting the accuracy of numerical solutions. By understanding the relationship between the mesh size, the order of basis functions, and the resulting error, practitioners can effectively design and apply FEM to solve complex Partial Differential Equations (PDEs) with confidence in the reliability of their results. Thus, the coercive property and A priori error estimation based on the L1 formula on a mesh in time and the Mamadu-Njoseh basis functions in space are investigated for a linearly distributed time-order fractional telegraph equation with restricted initial conditions. For this purpose, we constructed a mathematical proof of the coercive property for the fully discretized scheme. Also, we stated and proved a cardinal theorem for a priori error estimation of the approximate solution for the fully discretized scheme. We noticed the role of the restricted initial conditions imposed on the solution in the analysis of a priori error estimation.