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Weak form

Differential equations often demand more smoothness than simple elements can provide. By multiplying with a test function $w$ and integrating by parts, one derivative is shifted onto the test function. This gives the weak form, which requires less smoothness and fits naturally with the shape functions.

\int_\Omega w\left(-\dfrac{d}{dx}\left(k\dfrac{du}{dx}\right)\right)dx = \int_\Omega wf\,dx

strong form, before integrating by parts

Symbols

$w$	test function
$u$	unknown field (e.g. displacement)
$\Omega$	element domain

Example

Integrating by parts moves one derivative from $u$ to $w$ ,

so only first derivatives of both are needed, not second derivatives of $u$ .

The weak form is therefore less strict than the original (strong) differential equation.

Practise weak form and shape functions for free →

← Shape functions · Gauss quadrature →

Part of Finite Element Method: Weak form and shape functions.