Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, an equation is no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". This is equivalent to formulating the problem to require a solution in the sense of a distribution.
Contents
- General concept
- Example 1 linear system of equations
- Example 2 Poissons equation
- The LaxMilgram theorem
- Application to example 1
- Application to example 2
- References
We introduce weak formulations by a few examples and present the main theorem for the solution, the Lax–Milgram theorem.
General concept
Let
where
Calculus of variations tells us that this is equivalent to finding
Here, we call
We bring this into the generic form of a weak formulation, namely, find
by defining the bilinear form
Since this is very abstract, let us follow this by some examples.
Example 1: linear system of equations
Now, let
involves finding
where
Since
Actually, expanding
where
The bilinear form associated to this weak formulation is
Example 2: Poisson's equation
Our aim is to solve Poisson's equation
on a domain
to derive our weak formulation. Then, testing with differentiable functions
We can make the left side of this equation more symmetric by integration by parts using Green's identity and assuming that
This is what is usually called the weak formulation of Poisson's equation; what's missing is the space
We obtain the generic form by assigning
and
The Lax–Milgram theorem
This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the bilinear form. It is not the most general form.
Let
- bounded:
| a ( u , v ) | ≤ C ∥ u ∥ ∥ v ∥ and - coercive:
a ( u , u ) ≥ c ∥ u ∥ 2 .
Then, for any
and it holds
Application to example 1
Here, application of the Lax–Milgram theorem is definitely a stronger result than is needed, but we still can use it and give this problem the same structure as the others have.
Additionally, we get the estimate
where
Application to example 2
Here, as we mentioned above, we choose
where the norm on the right is the
Therefore, for any