On the Discretization Time-Step in the Finite Element Theta-Method of the Discrete Heat Equation

Abstract

In this paper the numerical solution of the one dimensional heat conduction equation is investigated, by applying Dirichlet boundary condition at the left hand side and Neumann boundary condition was applied at the right hand side. To the discretization in space, we apply the linear finite element method and for the time discretization the well-known theta-method. The aim of the work is to derive an adequate numerical solution for the homogenous initial condition by this approach. We theoretically analyze the possible choice of the time-discretization step-size and establish the interval where the discrete model is reliable to the original physical phenomenon.

As the discrete model, we arrive at the task of the one-step iterative method. We point out that there is a need to obtain both lower and upper bounds of the time-step size to preserve the qualitative properties of the real physical solution. The main results of the work is to determine the interval for the time-step size to be used in this special finite element method and analyze the main qualitative characterstics of the model.