Introduction to the Weak Form
The general form and coefficient form PDEs in equations Equation 16-1 and Equation 16-2 specify PDEs in a strong form, in the sense that they, in principle, require the PDE to be satisfied at every point in the geometry. And for this to be possible, all terms must be sufficiently continuous for derivatives and well-defined pointwise values to exist. In many cases, the natural phenomena a PDE intends to model are best described as discontinuous and may also contain source terms that are only defined as a total over a small region, without a well-defined pointwise value.
In these situations, a weak equation turns out to be a better model of physics than can be provided by the more commonly used strong form PDEs. In addition, the weak form is particularly suitable for discretization and numerical solution using the finite element method. One reason for this is precisely the lower continuity requirement on the solution, which only needs to be sufficiently smooth on each mesh element separately.
Example: Conversion from General Form to Weak Form
As an example, consider the general form presented in Equation 16-1, in particular the stationary form of the domain equation:
Assuming a single dependent variable u, introduce a corresponding arbitrary test function υ. Multiply the equation by this test function and integrate over the domain:
(16-5)
This integral equation is clearly a weaker statement than the original equation, in particular when Equation 16-5 is required to hold only for all test functions υ from a limited class of functions. In the finite element method, the test functions υ (and also solution u) are usually limited to the set of piecewise polynomials of a given order on each mesh element.
This polynomial can also be written as a sum of individual shape functions. Therefore, the original strong form PDE is transformed into a weak form equation, which must only be satisfied in a local integral sense over each shape function. When you increase the number of shape functions by refining the mesh or increasing the polynomial order, you simultaneously decrease the space of solutions u that can possibly satisfy Equation 16-5. Therefore, well-posed and consistent finite element formulations converge toward the single solution u that satisfies the original strong form PDE.
To further simplify the solution of Equation 16-5, the left-hand side integral can be integrated by parts, using Gauss law:
(16-6)
This has two main advantages. First of all, it reduces the maximum order of spatial derivatives. If Γ is a function of the gradient of u, for example Γ = −c∇u−αu+γ as in the coefficient form PDE, the transformed weak equation now contains only first-order derivatives compared to second-order derivatives in the original strong form PDE. Secondly, it makes it clear what the natural boundary condition is for this equation. The second integral on the left-hand side disappears if the normal component of Γ vanishes on the boundary. Alternately, if the value of the normal component is known, for example such that
(16-7)
on δΩ, this value can be inserted as a boundary condition into the weak form equation, which then becomes
(16-8)
This final weak formulation of the standard general form PDE therefore also explains why the Neumann boundary condition on the second line of Equation 16-1 looks the way it does.