Let
A basis of

The graph of
For some purposes piecewise constant function are more convenient then piecewise linear ones, therefore, we introduce a new function space on
For every edge
If we connect the center of the edges with the center of mass the triangle is divided into three parts with equal area. Now we can assign to each vertex

The domain
For the area of
The space of piecewise constant functions is defined in the following way:
Also on

Graph of
There is a similar construction that assigns to each vertex an face and works purely combinatorial: For a triangulated

A piece of an triangulated surface
Since in general it is not easy to compute the area of the dual faces with the metric of the original one, we used the above construction of the
On the same way we define an scalarproduct on
The matrix
This gives us a new scalar product on
Note that the basis
Now we have two scalar products on
Our aim is to define a discrete laplace operator
On a smooth surface
This identity should also be true in the discrete case and will serve us as definition for the discrete laplacian. We already defined the bilinear operator
and discussed the relation between quadratic forms, symmetric bilinear maps and maps between dual spaces ( lecture: dirichlet energy 2 ).
In the finite case the quadratic form, bilinear form and map between the dual spaces can all be represented by the same matrix
For
This gives us one laplace operator for each scalarproduct on
In general it is complicated and expensive to compute
One famous problem form physics and geometry is the Poisson problem:
Given
For Poisson problems both laplacians can be used since:
Example: Suppose
and for the potential
in order to determine the potential for a given density.
In this setting piecewise constant functions are much more convenient to describe the charge density then piecewise linear ones. Therefore we use