Weak solution

A function that satisfies a differential equation through an integral identity, often after derivatives are transferred to test functions, rather than through pointwise derivatives.

Test function: A smooth function, usually with compact support, used to probe a distribution or formulate an integral identity. Weak formulations test an equation against these functions instead of evaluating it pointwise.
Integration by parts: An identity that transfers differentiation between factors in an integral while producing boundary terms. It moves derivatives from a possibly rough solution onto a smooth test function.
Poisson equation: The elliptic equation stating that a function's Laplacian equals a prescribed source. Its integral formulation produces a standard model for weak solutions and energy methods.
Classical solution: A sufficiently differentiable function that satisfies a differential equation pointwise. A weak solution may exist when no function satisfies the equation with the required classical derivatives.
Distributional derivative: A derivative defined by transferring differentiation onto smooth test functions inside an integral. It lets weak solutions encode derivatives that may not exist pointwise.
Variational formulation: A formulation of a differential equation as a condition on integrals or on stationary values of a functional. Many weak equations arise by integrating the differential equation against test functions.
Finite element method: A numerical method that approximates a differential equation using piecewise-defined functions on a mesh. Finite element methods typically discretize a weak formulation rather than the pointwise equation.
Strong solution: A solution with enough regularity for the differential equation to hold almost everywhere in a pointwise sense. Strong solutions impose more derivative regularity than weak formulations alone require.
Lebesgue integral: An integral defined using measure and approximation by simple functions, allowing broad classes of measurable functions to be integrated. Weak formulations rely on integrals that remain meaningful for nonsmooth functions.
Boundary condition: A constraint imposed on a differential equation's solution at the boundary of its domain. Boundary terms in the weak identity determine how boundary data enter the problem.

Linked from 13 pages