Numerical Integration in Multiple Dimensions

Abstract

Galerkin methods require the evaluation of integrals of the type

$$\displaystyle A=\int _{\varOmega } f(\mathbf {x}) d\varOmega \, \, \mathrm {and} \, \, B=\int _{\varGamma } f(\mathbf {x}) d\varGamma $$

where Ω is the domain that we wish to integrate within and Γ are its boundaries. We saw in Chs. 5 and 6 that, in one dimension, A is a line integral while B denotes a jump value. In two dimensions, A is an area integral while B is a line integral. Finally, in three dimensions, A is a volume integral and B is a surface area integral. For brevity we always refer to A as a volume integral and B as either a boundary or flux integral.