Stiffness Matrix - Practical Assembly of The Stiffness Matrix

Practical Assembly of The Stiffness Matrix

In order to implement the finite element method on a computer, one must first choose a set of basis functions and then compute the integrals defining the stiffness matrix. 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 basis functions are then chosen to be polynomials of some order within each element, and continuous across element boundaries. The simplest choices are piecewise linear for triangular elements and piecewise bilinear for rectangular elements.

The element stiffness matrix A for element T_k is the matrix

The element stiffness matrix is zero for most values of i and j, for which the corresponding basis functions are zero within T_k. The full stiffness matrix A is the sum of the element stiffness matrices. In particular, for basis functions that are only supported locally, the stiffness matrix is sparse.

For many standard choices of basis functions, i.e. piecewise linear basis functions on triangles, there are simple formulas for the element stiffness matrices. For example, for piecewise linear elements, consider a triangle with vertices x₁, x₂, x₃, and define the 2×3 matrix

Then the element stiffness matrix is

When the differential equation is more complicated, say by having an inhomogeneous diffusion coefficient, the integral defining the element stiffness matrix can be evaluated by Gaussian quadrature.