1D MLSM and FDM comparison

Different numerical approaches to solving a Dirichlet or Neumann problem

<math> \begin{align*}

 \text{Dirichlet} && \text{Neumann} \\
 f(x) &= 2x^2+5 \text{ on } (0, 1) &   f(x) &= 2x^2+5 \text{ on } (0, 1)  \\
 f(0) &= 1 & f'(0) &= 1 \\
 f(1) &= 1 & f(1) &= 1 \\
 f(x) &= \frac{1}{6} \left(x^4+15 x^2-16 x+6\right) & f(x) &= \frac{1}{6} \left(x^4+15 x^2+6 x-16\right) 

\end{align*} </math>

were analysed. Theoretically, FDM and MLSM should match completely. This is practivaly demonstrated up to certain discretization level.

The interval <math>[0, 1]</math> was always discretized uniformly using $N$ nodes, <math>x_i = a+i h, h = (b-a)/N</math>.

Dirichlet case

Precision and execution time are summarised in graphs below.