Finite element analysis for microscale heat equation with Neumann boundary conditions

Document Type : Research Article

Authors

• M.H. Hashim
• A.J. Harfash

Department of Mathematics, College of Sciences, University of Basrah, Basrah, Iraq.

10.22067/ijnao.2024.87084.1403

Abstract

In this paper, we explore the numerical analysis of the microscale heat equation. We present the characteristics of numerical solutions obtained through both semi- and fully-discrete linear finite element methods. We establish a priori estimates and error bounds for both semi-discrete and fully-discrete finite element approximations. Additionally, the existence and uniqueness of the semi-discrete and fully-discrete finite element ap-proximations have been confirmed. The study explores error bounds in various spaces, comparing the semi-discrete to the exact solutions, the semi-discrete against the fully-discrete solutions, and the fully-discrete solutions with the exact ones. A practical algorithm is introduced to address the sys-tem emerging from the fully-discrete finite element approximation at every time step. Additionally, the paper presents numerical error calculations to further demonstrate and validate the results.

Keywords

• Finite element
• Microscale heat equation
• Convergence
• Weak solution

Main Subjects

• Numerical methods for differential equations and its applications

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