---
title: "Compare weighted residual methods"
subject: "Finite Element Method"
chapter: 4
key: "semester_8/finite_element_method:weighted-residual-methods-compare"
source: https://www.bctengineer.com/q/semester_8/finite_element_method/weighted-residual-methods-compare
site: BCT Engineering
---

# Compare weighted residual methods

Finite Element Method · Chapter 4: Continuum Problems

- Asked in: year not given, 6 marks
- Video: [Weighted Residual (1/5): Intro & Process](https://www.youtube.com/watch?v=Ym7FDhvf4Kk) — Mike Foster
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/finite_element_method/weighted-residual-methods-compare
- Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-4

## Question

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

## Answer

#### Method of weighted residuals (MWR)

For a differential equation $\mathcal{D}(u) + f = 0$ on a domain $\Omega$, an approximate solution $\tilde{u} = \sum a_i\phi_i(x)$ does not satisfy the equation exactly, leaving a **residual**
$$
R(x, a_i) = \mathcal{D}(\tilde{u}) + f \neq 0
$$
The MWR makes the residual small by forcing a weighted integral to be zero:
$$
\int_\Omega W_i(x)\,R(x, a_i)\,d\Omega = 0,\qquad i = 1, \dots, n
$$
This gives $n$ equations for the $n$ unknowns $a_i$. The choice of the weight function $W_i$ defines the method.

#### Comparison

| Method | Weight function $W_i$ | Condition imposed |
|---|---|---|
| Point collocation | Dirac delta, $\delta(x - x_i)$ | $R = 0$ at $n$ chosen points |
| Subdomain | $1$ in subdomain $i$, $0$ elsewhere | $\int_{\Omega_i} R\,d\Omega = 0$ over each subregion |
| Least squares | $\partial R/\partial a_i$ | Minimises $\int R^2d\Omega$ |
| Galerkin | $\phi_i$ (same as trial function) | $\int \phi_i R\,d\Omega = 0$ |

#### 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the residual in the method of weighted residuals?
   A. The error left when the approximate solution is substituted in the differential equation, R = D(u~) + f
   B. The load on the structure
   C. The boundary reaction
   D. The exact solution
   Answer: A. The approximate solution does not satisfy the equation exactly, leaving a residual R which the method makes small by a weighted integral. (Key point: Residual)

2. What condition determines the n unknowns a_i?
   A. R = 0 everywhere
   B. Integral of W_i R dOmega = 0 for i = 1..n
   C. Sum of a_i = 0
   D. Integral of R dOmega = 1
   Answer: B. Forcing the weighted integral to vanish gives n equations; the weight function defines the method. (Key point: Condition)

3. What weight function does point collocation use?
   A. The trial function
   B. dR/da_i
   C. Dirac delta
   D. 1 in a subdomain
   Answer: C. A delta weight forces R = 0 at n chosen points. (Key point: Collocation)

4. What weight does the least squares method use?
   A. phi_i
   B. Dirac delta
   C. 1
   D. dR/da_i
   Answer: D. W_i = dR/da_i, which minimises the integral of R^2. (Key point: Least squares)

## Related in Finite Element Method

- [Galerkin and collocation solution of an ODE](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Agalerkin-collocation-ode) — Finite Element Method, chapter 4

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