ABSTRACT
This chapter considers the techniques for developing continuous approximate solutions based on the method of weighted residuals. By providing approximations to the primary dependent variables ϕ at all points within the domain Ω and along its boundary Γ, such solutions overcome the fundamental shortcoming of discrete approaches such as the finite difference method. However, continuous approximate solutions require the analyst to choose a set of suitable trial functions, and a set of corresponding weighting functions, for use in the method of weighted residuals. In addition, both of these sets of functions must apply to the entire domain. For geometrically simple domains such as lines, rectangles, right prisms, and so on, this fact poses no difficulties. However, in the rase of domains with curved boundaries, cut-outs, material inhomogeneities, inclusions, and so on, the choice of suitable trial solutions becomes substantially more difficult.
