3 Weighted Residual Methods
3.1 Start with a residual
Suppose a linear differential equation is \(\mathcal L u=f\) and an approximate field is \(u_h=\sum_j N_j a_j\). The pointwise residual is
\[\mathcal R(u_h)=\mathcal L u_h-f.\]
A weighted-residual method chooses test functions \(w_i\) and enforces
\[\int_\Omega w_i\,\mathcal R(u_h)\,d\Omega=0.\]
The residual need not vanish at every point; instead, it is orthogonal to a selected test space in a chosen integral sense.
3.2 Common families
- Galerkin: test functions are drawn from the trial space.
- Petrov–Galerkin: trial and test spaces may differ.
- Least-squares: seeks to minimize an integral norm of a residual, subject to an appropriate formulation.
- Collocation: enforces residual conditions at selected points rather than through the displayed integral form.
3.3 Integration by parts
When a second-order operator is integrated by parts, one derivative may move from the trial field to a test function. Boundary terms appear and must be treated according to the boundary conditions.
Exercise: Derive the weak form of \(-u''=f\) on \((0,1)\) with \(u(0)=u(1)=0\).
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.