Re: Irregularly gridded Discrete Laplacian Operator

[Geordie McBain]

> Is there a "correct" definition for irregularly gridded discrete
> Laplacian operator? Is there any reason to prefer either of the above
> definitions?

I'm not sure about a correct definition, but one definition is a finite
difference approximation that is exact for polynomials of such and such
a degree.  This is what in my work I use, directly, to compute finite
difference coefficients; i.e. I solve the linear system of equations
that specifies that the multiplication of the row of N coefficients by
the N columns of N ordinates of N test functions (an N*N matrix) at the
N abscissae gives the N derivatives (of whatever order) of the test
functions at the target point.  I attach my implementation, in case it's
of interest.  To get the second derivative operator; e.g.
%<---
octave:11> x = sort (rand (6, 1)); x'
ans =

   0.046615   0.325824   0.712196   0.769261   0.852244   0.969730

octave:12> full (finidiff_matrix (x, 2))
ans =

     0.00000     0.00000     0.00000     0.00000     0.00000     0.00000
    10.76213   -18.53932     7.77719     0.00000     0.00000     0.00000
     0.00000    11.67327   -90.71033    79.03706     0.00000     0.00000
     0.00000     0.00000   250.25795  -422.35298   172.09502     0.00000
     0.00000     0.00000     0.00000   120.22535  -205.14261    84.91725
     0.00000     0.00000     0.00000     0.00000     0.00000     0.00000
%<---
Although solving a dense linear system might seem expensive, it's only O
(n), where n is the number of grid points, in the context of solving a
differential equation on the grid.  Such an important special case as
the Laplacian, however, is probably worth hard-coding.

Best wishes, and keep up the good work.
Geordie McBain

finidiff.m
Description: Text Data

finidiff_matrix.m
Description: Text Data