finite element analysis for microscale heat equation with neumann boundary conditions

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.