One can write a general differential equation as:
An example, which we will often consider is
to model e.g. heat conduction with a space-dependent heat conduction coefficient and a heat production term . (Notice the “ex” when we are talking about a particular example.)
This is pretty technical, so please stop reading here if you are already lost!
On top of that we of course need boundary conditions, Neumann, Dirichlet, Robin… In our case we will consider
Three steps to the weak form
Step 1 – weighted residual
Cast the problem as in the method of weighted residuals (that article is pretty similar to this one up to here). In the end, we have:
where we have approximation functions (aka solution functions) and weight functions (aka test functions) .
Step 2 – weak form
Now, only if the involves derivatives of even order, we can evenly distribute the derivatives between and , thanks to integration by parts. In this instance,
(notice we write for any of the weight functions) is converted into
This is the weak form of the original differential equation. It is weaker in the sense that, clearly, less derivatives of appear than in the original equation (in this instance, we go from second to only first). This translates into less derivatives for the solution functions, too.
Step 3 – boundary conditions
We must not forget the boundary conditions. In the weak form, we can clearly spot the boundary term, the last one:
In general, we may encounter natural and essential boundary conditions (NBCs and EBCs, resp.).
NBCs are the ones that relate to and its derivatives at the boundaries. In this instance, we car readily identify as the term that multiplies . This term is often of physical importance: in this case we can identify it with the heat flux at the borders, , where the sign depends on the border: on the left, , we have a plus for heat influx; at the right we have a minus for the same reason. All together, we may write
EBCs are the ones that relate to or its derivatives at the boundaries. This term is, however, deduced from the same as the previous one: one must look at the way in which appears: this will be the way appears too. In this instance, appears simply alone, so the EBCs will refer to only (not its derivatives). (I know this is confusing, read twice!)
In general, NBCs and EBCs come in pairs. Both carry physical information, on quantities that are usually mutually conjugate in some sense. Only one of the two must be specified at each boundary, though. In this instance
- The EBC at the left boundary, trivially.
- The NBC at the right boundary, trivially.
- The NBC at the left boundary, not trivially.
- An Introduction to the Finite Element Method, J. Reddy. McGraw-Hill; 3 edition (January 11, 2005). ISBN-10: 0072466855. At amazon (167 bucks!).
- Petrov-Galerkin_method at wikipedia. (Not to be trusted so much!)
- Method of mean weighted residuals at wikipedia
- Galerkin method at wikipedia
- The method of weighted residuals
- The Rayleigh-Ritz method (to be written!)