Weak Formulation

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 -del ^2u=f on a domain Omega, with u=0 on the boundary, has weak formulation

for every test function v that vanishes on the boundary. The unknown u is sought in the corresponding Sobolev space. Weak formulations provide the basis for variational methods and the finite element method.

See also

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References

Cite this as:

Weisstein, Eric W. "Weak Formulation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeakFormulation.html

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