Study guide: Introduction to finite element methods

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Variable coefficient; Galerkin principle

$R = -\frac{d}{dx}\left( a\frac{du}{dx}\right) -f$

Galerkin's method: $(R, v) = 0,\quad \forall v\in V$

or with integrals: $\int_{\Omega} \left(\frac{d}{dx}\left( \dfc\frac{du}{dx}\right) -f\right)v \dx = 0,\quad \forall v\in V$