LTCC Course Variational and Computational Methods for PDEs

∆u = ∂2u ∂x1 + ∂2u ∂x2 where x = (x1, x2) ∈ R2 are Cartesian coordinates. Even though the Poisson equation looks very special it is an important model case representing several problems from physics and engineering, e.g. electrostatics, stationary heat transfer and other diffusion problems. Variations of the techniques we will study apply to a wide class of second order so-called elliptic problems. It is known that there are cases where no classical (i.e. twice continuously differentiable) solution of (1.1) exists. In order to deal with a uniquely solvable problem one therefore derives a weak formulation. It is convenient to write the Laplace operator in the following form: