Prototype Two-Dimensional (2D) Problem: Variational Formulations

In this chapter, we will extend to higher dimensions the study of the prototype 1D BVP. This will lead to a new BVP based on a classical second order partial differential equation, the generalized Poisson’s equation. As a concrete physical application, we will consider the case of steady-state heat conduction in a 2D domain with a closed boundary. We say this is a steady-state problem because we wish to examine the situation where the system has reached a final state in which the temperature does not vary in time, just spatially. We define the 2D model problem below and also provide an appropriate formula for integration by parts which we will need for deriving the variational formulation of the BVP, but we will omit some of the technical mathematical details.