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Compare weighted residual methods

Short answer

MWR sets ∫ Wᵢ R dΩ=0. Point collocation uses Dirac delta weights (R=0 at chosen points). Subdomain uses unit weight over subregions. Least squares minimises ∫R² dΩ (weights ∂R/∂aᵢ). Galerkin uses trial functions as weights, yielding symmetric matrices for self-adjoint problems; it is the basis of most FEM formulations.

Explain the method of weighted residuals. Differentiate between point collocation, subdomain, least squares and Galerkin methods.

Method of weighted residuals (MWR)

For a differential equation D(u)+f=0\mathcal{D}(u) + f = 0 on a domain Ω\Omega, an approximate solution u~=∑aiϕi(x)\tilde{u} = \sum a_i\phi_i(x) does not satisfy the equation exactly, leaving a residual

R(x,ai)=D(u~)+f≠0R(x, a_i) = \mathcal{D}(\tilde{u}) + f \neq 0

The MWR makes the residual small by forcing a weighted integral to be zero:

∫ΩWi(x) R(x,ai) dΩ=0,i=1,…,n\int_\Omega W_i(x)\,R(x, a_i)\,d\Omega = 0,\qquad i = 1, \dots, n

This gives nn equations for the nn unknowns aia_i. The choice of the weight function WiW_i defines the method.

Comparison

MethodWeight function WiW_iCondition imposed
Point collocationDirac delta, δ(x−xi)\delta(x - x_i)R=0R = 0 at nn chosen points
Subdomain11 in subdomain ii, 00 elsewhere∫ΩiR dΩ=0\int_{\Omega_i} R\,d\Omega = 0 over each subregion
Least squares∂R/∂ai\partial R/\partial a_iMinimises ∫R2dΩ\int R^2d\Omega
Galerkinϕi\phi_i (same as trial function)∫ϕiR dΩ=0\int \phi_i R\,d\Omega = 0

Remarks

  • Collocation is simplest but depends on the choice of points.
  • Least squares always gives a symmetric matrix but needs higher-order derivatives, so the algebra is heavy.
  • Galerkin gives a symmetric matrix for self-adjoint problems and equals the Ritz method when a functional exists. It is the basis of most finite element formulations.
  • All the methods become exact if the residual is zero everywhere.

✓ Answer checked by Claude Opus 5.5 (high effort)

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