To discretize this equation by the finite element method, one chooses a set of basis functions {φ1, …, φn} defined on Ω which also vanish on the boundary.
the coefficients ui are determined by the linear system Au = F. The stiffness matrix is symmetric, i.e. Aij = Aji, so all its eigenvalues are real.
Note that the stiffness matrix will be different depending on the computational grid used for the domain and what type of finite element is used.
Determining the stiffness matrix for other PDEs follows essentially the same procedure, but it can be complicated by the choice of boundary conditions.
We impose the Robin boundary condition where νk is the component of the unit outward normal vector ν in the k-th direction.
Usually, the domain Ω is discretized by some form of mesh generation, wherein it is divided into non-overlapping triangles or quadrilaterals, which are generally referred to as elements.
The condition number of the stiffness matrix depends strongly on the quality of the numerical grid.
In particular, triangles with small angles in the finite element mesh induce large eigenvalues of the stiffness matrix, degrading the solution quality.