Partial Differential Equations – PDE is called “strong form” because the relationship MUST satisfy at every mathematical point in the domain.
A “weak form” means that the relationship (in integral form) is only satisfied in overall sense. In another word, “it is only satisfied in an integral (sum) sense, it is not a requirement that every point in the domain MUST obey”
The strong form of a differential equation is just that: the (partial) differential equation itself. Evaluating the PDE requires being able to get all the associated derivatives. It is satisfied pointwise at every point in a body, and is usually stated as D[u] = 0, where D is some partial differential operator. In this case, I am using u as the displacement. It may be more appropriate to look at, say, Cauchy stress (s) instead of displacement. Then the strong form might be something like D[s,u] = div[s] – ru,tt= 0.
The weak form is obtained by multiplying the PDE by an arbitrary weighting function of (in most cases) the spatial variables, then integrating the result over the domain. One then requires that the result is zero for all choices of such functions. An integration by parts is performed, leading to differentiability requirements on the weighting function, but relaxing, or “weakening” the requirements on the field described by the PDE. Once we start setting requirements on these functions, we “weaken” the form even more, but often provide a basis for expressing the approximate solution.
Q1. Why do we multiply the PDE by the weighting function?
Q2. How do we choose the weighting function?
The “why” is to reduce differentiability requirements on our approximate solution.
The “how” is whatever works. If you know of a particular set of functions that work well in your geometry (say Bessel functions for axisymmetric problems), you can use these. We often use arbitrary combinations of nodal basis functions (compact support, C1, finite domains). If we use the same basis for our weighting functions as we use to represent our primary variable field(s), e.g. displacement, we are using a Galerkin method.
By Matt Lewis
Los Alamos, New Mexico
This excerpt is taken from IMECHANICA Forum.
- Weak Derivative – Wikipedia
- Weak Formulation – Wikipedia
- Weak Solution – Wikipedia