The Method of Weighted Residuals (MWR)

Consider the well-posed differential equation

$$\mathcal{L} u(\mathbf{x})=f(\mathbf{x}) , \quad \mathbf{x}\in\Omega\subseteq\mathbb{R}^n$$

with boundary conditions on $\partial \Omega$. We will approximate the solution $u(\mathbf{x})$ as

$$v(\mathbf{x})=\sum_{j=1}^N c_j\phi_j(\mathbf{x})$$

where $\{\phi_j\}_{j=1}^N$ span the ``trial space''. Hence, we denote $v(\mathbf{x})$ as the ``trial functions''. The goal in MWR is the determination of the $N$ scalars $\{c_j\}_{j=1}^N$.

This is done in MWR by minimizing

$$r(\mathbf{x})\equiv\mathcal{L}v(\mathbf{x_0}-f(\mathbf{x})$$

where $r(\mathbf{x})$ is the ``residual''. We do so by attemping to find the coefficients that drive the weighted average

$$\int_{\Omega}r(\mathbf{x})w_i(\mathbf{x}) \mathrm{d}\mathbf{x}=0 \qquad \forall i,$$

where $\{w_i\}_{i=1}^M$ are the ``test functions'' or weights. The number of weight functions and $N$ are related as shown below.

A good choice of basis functions are the Lagrange poynomials. To be specific, let $n=1$ and $\{x_j\}_{j=1}^N$ be the set of $N$ ``nodes'' on some intreval $[x_0,x_f]\equiv\Omega$, they are distinct but not necessarily evenly spaced. Here $x_0$ and $x_f$ are the boundary points $\partial \Omega$.

An $(N-1)$$^{th}$ degree polynomial associalted with $x_j$, $j=1,\ldots, N$

$$\ell_j(x)=\prod_{i=1,i\neq j}^{N} \frac{x-x_i}{x_j-x_i}$$

forms a basis of $S_N$, the finite-dimensional linear space. As we saw in 475A

$$v_i\equiv v _(x_i)=\sum_{j=1}^N c_j \ell_j(x_i)=c_i \ell_i(x)=c_i,$$

since $\ell_j(x_i)=\delta_{ij}$. So we have

$$v(x)=\sum_{j=1}^N v_j\phi_j(x)$$

where the $\phi_j$ are Lagrange polynomials.

These $\ell_j(x)$ are nonzero over entire $\Omega$, except at a finite number of node points. Further, $\ell_j(x)\in C^{\infty}(\Omega)$. However, we may want to choose piecewise polynomials rather than global polynimials. This is beneficial for parallel computing where we want to minimze communication between processors.

The following sets of piecewise defined polynomials in $C^0(\Omega)$ are simple and thus quite popular.

• piecewise linear
  $$\phi_j(x)= \begin{cases} \frac{x-x_{j-1}}{x_j-x_{j-1}} \quad ... ... x_{j+1}\\ \quad 0 \qquad \qquad \text{otherwise} \end{cases}$$
  Download the piecewise linear polynomial interpolation matlab script.

• piecewise quadratic and still $C^0(\Omega)$

  $j$ odd

  $$\phi_j(x)= \begin{cases} \frac{(x-x_{j-1})(x-x{j-2})}{(x_j-x_... ... x_{j+2}\\ \quad 0 \qquad \qquad \text{otherwise} \end{cases}$$

  $j$ even

  $$\phi_j(x)= \begin{cases} \frac{(x-x_{j-1})(x_{j+1}-x)}{(x_j-x... ... x_{j+1}\\ \quad 0 \qquad \qquad \text{otherwise} \end{cases}$$
  Download the piecewise quadratic interpolation matlab script. Compare to the piecewise linear and the cubic interpolation cases (see below).
• for $C^1(\Omega)$ functions use piecewise Hermite polynimoals
  $$v(x)=\sum_{j=1}^N v_j\phi_{oj}(x)+\frac{ \mathrm{d}v_j}{ \mathrm{d} x}\phi_{1j}(x) .$$
  Here $v_j$ is the approximation $u$ at $x_j$, $\frac{ \mathrm{d}v_j}{ \mathrm{d}x}$ is the approximation of $\frac{ \mathrm{d}u}{ \mathrm{d}x}$ at $x_j$ and $\phi_{0j}$ and $\phi_{1j}$ are piecewise Hermite polynomials. Here are the cubic Hermite polynomials:
  $$\phi_{0j}(x)= \begin{cases} \frac{(x-x_{j-1})^2 [2(x_j-x)+(x_... ... x_{j+1}\\ \quad 0 \qquad \qquad \text{otherwise} \end{cases}$$
  $$\phi_{1j}(x)= \begin{cases} \frac{(x-x_{j-1})^2 (x-x_j)}{(x_j... ... x_{j+1}\\ \quad 0 \qquad \qquad \text{otherwise} \end{cases}$$

  with the properties

  $$\phi_{0j}(x_i)=\delta_{ij}\qquad \phi_{0j}(x_i)=0$$

  $$\frac{ \mathrm{d}\phi_{0j}}{ \mathrm{d}x}(x_i)=0 \qquad \frac{ \mathrm{d}\phi_{1j}}{ \mathrm{d}x}(x_i)=\delta_{ij} .$$

  
  

Download the piecewise cubic Hermite polynomial interpolation matlab script. Verify the properties of the interpolants and compare the results of the cubic interpolation to the quadratic and linear cases.

Recall that MWR goal is to minimize $r(x)$ by forcing it to zero in a weighted average sense over the domain $\Omega$. The most popular variants of MWR are

1. Subdomain Method
2. Collocation
3. Galerkin and Petrov-Galerkin

We will only present the first 2 by example.