Mohammad Asadzadeh - An Introduction to the Finite Element Method for Differential Equations

and the time rate (time derivative) of change of thermal energy in is:

Let denote the heat flux vector and denote the outward unit normal to the boundary , at the point . Then represents the flow of heat per unit cross‐sectional area per unit time crossing a surface element. Thus,

is the amount of heat per unit time flowing into across the boundary . Here, represents the element of surface area. The minus sign reflects the fact that if more heat flows out of the domain than in, the energy in decreases. Finally, in general, the heat production is determined by external sources that are independent of the temperature. In some cases, (such as an air conditioner controlled by a thermostat), it depends on temperature itself, but not on its derivatives. Hence, in the presence of a source (or sink), we denote the corresponding rate at which heat is produced per unit volume by so that the source term becomes

Now, the law of conservation of energy takes the form

(1.5.8)

Applying the Gauss divergence theorem to the integral over , we get

(1.5.9)

where denotes the divergence operator. In the sequel, we shall use the following simple result:

Let be a continuous function satisfying for every domain . Then .

Let us assume to the contrary that there exists a point where . Assume without loss of generality that . Since is continuous, there exists a domain (maybe very small) , containing , and an , such that , for all . Therefore, we have , which contradicts the assumption.

From ( 1.5.9), using Lemma 1.1, we conclude that

(1.5.10)

This is the basic form of our heat conduction law. The functions and are unknown and additional information of an empirical nature is needed to determine the equation for the temperature . First, for many materials, over a fairly wide but not too large temperature range, the function depends nearly linearly on , so that