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Concept
Weak formulation
Summary
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, equations or conditions are 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". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.
The Lax–Milgram theorem, named after Peter Lax and Arthur Milgram who proved it in 1954, provides weak formulations for certain systems on Hilbert spaces.
Let be a Banach space, its dual space, , and . Finding the solution of the equation
is equivalent to finding such that, for all ,
Here, is called a test vector or test function.
To bring this into the generic form of a weak formulation, find such that
by defining the bilinear form
Now, let and be a linear mapping. Then, the weak formulation of the equation
involves finding such that for all the following equation holds:
where denotes an inner product.
Since is a linear mapping, it is sufficient to test with basis vectors, and we get
Actually, expanding , we obtain the matrix form of the equation
where and .
The bilinear form associated to this weak formulation is
To solve Poisson's equation
on a domain with on its boundary, and to specify the solution space later, one can use the -scalar product
to derive the weak formulation. Then, testing with differentiable functions yields
The left side of this equation can be made more symmetric by integration by parts using Green's identity and assuming that on :
This is what is usually called the weak formulation of Poisson's equation. Functions in the solution space must be zero on the boundary, and have square-integrable derivatives. The appropriate space to satisfy these requirements is the Sobolev space of functions with weak derivatives in and with zero boundary conditions, so .
Official source
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