Version 1
: Received: 4 October 2024 / Approved: 7 October 2024 / Online: 7 October 2024 (17:34:13 CEST)
How to cite:
Osuntoki, J. Advanced Solutions to Boundary Value Problems: A Comparative Study of Rayleigh-Ritz and the Finite Element Method. Preprints2024, 2024100461. https://doi.org/10.20944/preprints202410.0461.v1
Osuntoki, J. Advanced Solutions to Boundary Value Problems: A Comparative Study of Rayleigh-Ritz and the Finite Element Method. Preprints 2024, 2024100461. https://doi.org/10.20944/preprints202410.0461.v1
Osuntoki, J. Advanced Solutions to Boundary Value Problems: A Comparative Study of Rayleigh-Ritz and the Finite Element Method. Preprints2024, 2024100461. https://doi.org/10.20944/preprints202410.0461.v1
APA Style
Osuntoki, J. (2024). Advanced Solutions to Boundary Value Problems: A Comparative Study of Rayleigh-Ritz and the Finite Element Method. Preprints. https://doi.org/10.20944/preprints202410.0461.v1
Chicago/Turabian Style
Osuntoki, J. 2024 "Advanced Solutions to Boundary Value Problems: A Comparative Study of Rayleigh-Ritz and the Finite Element Method" Preprints. https://doi.org/10.20944/preprints202410.0461.v1
Abstract
Purpose - This study investigates the solution of a boundary value problem using two numerical methods: the Rayleigh-Ritz method and the finite element method (FEM). The aim is to compare the performance of these methods and assess the reliability of FEM as an generalization of the Rayleigh-Ritz approach for more complex problems. Design/Methodology/Approach - The Rayleigh-Ritz method and the linear element formulation of the finite element method were employed to solve the boundary value problem. A detailed comparison of the results obtained from both methods was performed. Graphical illustrations were used to present the solutions, and potential sources of error were analyzed, including element and domain approximation errors, round-off errors, and the impact of using linear rather than quadratic elements in FEM. Findings - The solutions generated by both methods were found to be in close agreement, demonstrating that FEM is a viable alternative to the Rayleigh-Ritz method for solving boundary value problems. The minor discrepancies observed can be attributed to approximation errors and the choice of linear elements in the finite element analysis. Originality/Value - This work highlights the applicability and effectiveness of both the Rayleigh-Ritz method and FEM in solving boundary value problems. It underscores the finite element method’s flexibility, especially in handling more complex boundary conditions and geometries, and contributes to the understanding of the factors influencing the accuracy of numerical methods in structural analysis.
Keywords
Keywords: Partial Differential Equations; Boundary Value Problem; Rayleigh-Ritz Method: Finite Element Method (FEM); Numerical Analysis
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
