MWLS Approximation

Introduction to the WRNK method

The Weighted Moving Least Squares (WRNK) method (in English: Moving Weighted Least Squares MWLS) is a relatively new and advanced approximation method that is used, for example, in the meshless Galerkin method (BMG).

We will not analyze the mathematical background of this method, as it is quite complicated and frankly, not entirely necessary for learning its application.

Solution process for the 1D (one-dimensional) case

1. Taylor series expansion

We write the value of the function using the Taylor series expansion (in practice, most often Maclaurin, since we assume \(a=0\) whenever possible):

\[ \begin{aligned} f(x) &= f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots \\ &\quad + \frac{f^{(n)}(a)}{n!}(x-a)^n + \dots \\ &= \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n \end{aligned} \]

For \( a = 0 \):

\[ \begin{aligned} f(x) &= f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \dots \\ &= \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \end{aligned} \]

The degree of approximation determines the number of elements we examine, that is, for an approximation degree of \( p=1 \), we write the formula only up to and including the first derivative.

2. Error function

We write the error function:

\[ \begin{aligned} J &= (\tilde{u}_1 - u_1)^2 \cdot \left( \frac{1}{h^{p+1-s}} \right)^2 \\ &\quad + (\tilde{u}_1 - u'_1)^2 \cdot \left( \frac{1}{h^{p+1-s}} \right)^2 \\ &\quad + (\tilde{u}_2 - u_2)^2 \cdot \left( \frac{1}{h^{p+1-s}} \right)^2 + \dots \end{aligned} \]

For all elements that are known.

3. Minimization of the error function

We minimize the error function with respect to the examined value, e.g., \( u_1 \):

\[ \begin{aligned} \frac{d}{du_1} J = 0 \end{aligned} \]

And based on this, we calculate the sought value.