Start page of documentation

Download the code

Documentation: PDF, PS

Discretisation in a rectangle

Over $\Omega^*$ a regular grid with mesh widthes $h_x$ and $h_y$ in $x$ and $y$ directions respectively is imposed. In code they are denoted by hx and hy. $n_x$ and $n_y$, in code nx and ny (ASSEMBLE/problem_setup.m) are the number of grid points in $x$ and $y$ directions. Important is to note that number of grid points should be counted in Matlab convention, it is, starting from 1 and not from 0.

Due to the simplified geometry pre-processor which is used in the online version of the code, the parameters $n_x$ and $n_y$ have to be selected such that $h_x$ and $h_y$ are equal. This restriction can be removed without any further changes if the geometry pre-processor is replaced.

The $\Delta$ operator of (1) is discretised in $\Omega^*$ by standard central finite differences:

$$\Delta u\approx \frac{1}{h_x^2}\left( u_{i+1,j}-2u_{i,j}+u_{i-1,j}\right) +\frac{1}{h_y^2}\left( u_{i,j+1}-2u_{i,j}+u_{i,j-1}\right) .$$ (2)

The resulting standard finite difference matrix is denoted by $\mathbf{A}$, is stored in variable A and computed by function DIFFOP/sysmatrix_poisson.m.