View source for Analysis of MLSM performance

For starters, we can solve simple Diffusion equation
$ \nabla^2 u = \frac{\partial u}{\partial t} $.

We solved the equation on a square $\Omega = [0, a] \times [0, a]$ with
Dirichlet boundary conditions $ \left. u\right|_{\partial \Omega} = 0 $ and
initial state $ u(t = 0) = 1$.

An analytical solution for this domain is known, and we use it to evaluate or
own solution. 
\begin{equation} 
u(\vec{p}, t) = \sum_{\substack{n=1 \\ n \text{
odd}}}^\infty\sum_{\substack{m=1 \\ m \text{ odd}}}^\infty \frac{1}{\pi^2}
\frac{16 a^2}{nm} \sin\left(\frac{\pi n}{a}p_x\right) \sin\left(\frac{\pi
m}{a}p_y\right) e^{-\frac{\pi^2 (n^2+m^2)}{a^2}t} 
\end{equation}
Because the solution is
given in the series form, we only compare to the finite approximation, summing
to $N = 100$ instead of infinity.

A picture of our solution (with smaller and larger node density):

[[File:square_heat.png|400px|center]]

=Analysis of our method=

==Convergence with respect to number of nodes==

[[File:node_convergence.png|400px|center]]

We tested the method with a fixed time step of $ \Delta t = 1\cdot 10^{-5}$
on a unit square ($a = 1$).  Monomial basis of 6 monomials was used and 12
closest nodes counted as support for each node.  After more than 250 nodes of
discretization in each dimension the method diverges, which is expected. The
stability criterion for diffusion equation in two dimensions is $\Delta t \leq
\frac{1}{4} \Delta x^2$, where $\Delta x$ is the spatial discretization
step in one dimension. In our case, at 250 nodes per side, the right hand side
yields $\frac{1}{4}\cdot\frac{1}{250}\cdot\frac{1}{250} = 4\times 10^{-6}$,
so our method is stable within the expected region.