A weak formulation rewrites a differential equation as an integral identity against a class of test functions, transferring derivatives from the unknown function by integration by parts. Solutions of the resulting identity are called weak solutions and may have fewer classical derivatives than the original differential equation requires.
For example, the homogeneous Dirichlet Poisson problem 
on a domain 
, with 
on the boundary, has weak formulation
 |
for every test function 
that vanishes on the boundary. The unknown 
is sought in the corresponding Sobolev
space. Weak formulations provide the basis for variational methods and the finite element method.
