Weak Formulation of PDEs
The weak form recasts a differential equation as an integral identity that must hold against all test functions, admitting less regular solutions.
Strong versus weak
A partial differential equation stated pointwise is the strong form; it demands that the solution be differentiable enough for every derivative to exist everywhere. Many physically meaningful solutions do not meet that bar: fields with kinks at material interfaces, or discontinuities across shocks. The weak formulation relaxes the requirement by asking the equation to hold only in an averaged sense.
To derive it, multiply the equation by an arbitrary test function v, integrate over the domain, and use integration by parts to shift derivatives from the unknown onto v. This lowers the differentiability the solution must have and moves boundary terms into the formulation explicitly.
Function spaces
The natural setting is a Sobolev space, typically H^1, whose members have square-integrable first derivatives. A weak solution lives in this space and satisfies the integral identity for every admissible test function. The Lax-Milgram theorem gives conditions (boundedness and coercivity of the bilinear form) under which a unique weak solution exists.
Why it matters for discretization
The weak form is the foundation of the finite element and Galerkin methods: restricting the trial and test spaces to finite-dimensional subspaces turns the integral identity into a solvable linear system. Natural boundary conditions (fluxes) appear automatically from the boundary integrals, while essential conditions (prescribed values) are built into the trial space.
- Test functions probe the equation in every admissible direction
- Integration by parts reduces required regularity of the solution
- Coercivity and continuity ensure well-posedness
- Natural boundary conditions emerge without extra equations
Understanding the weak form clarifies why finite element solutions conserve certain quantities and why their error can be bounded in energy norms tied directly to the physics.