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Concept
Stiffness matrix
Summary
In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.
For simplicity, we will first consider the Poisson problem
on some domain Ω, subject to the boundary condition u = 0 on the boundary of Ω. To discretize this equation by the finite element method, one chooses a set of basis functions {φ_1, ..., φ_n} defined on Ω which also vanish on the boundary. One then approximates
The coefficients u_1, u_2, ..., u_n are determined so that the error in the approximation is orthogonal to each basis function φ_i:
The stiffness matrix is the n-element square matrix A defined by
By defining the vector F with components the coefficients u_i are determined by the linear system Au = F. The stiffness matrix is symmetric, i.e. A_ij = A_ji, so all its eigenvalues are real. Moreover, it is a strictly positive-definite matrix, so that the system Au = F always has a unique solution. (For other problems, these nice properties will be lost.)
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. For example, the stiffness matrix when piecewise quadratic finite elements are used will have more degrees of freedom than piecewise linear elements.
Determining the stiffness matrix for other PDEs follows essentially the same procedure, but it can be complicated by the choice of boundary conditions. As a more complex example, consider the elliptic equation
where is a positive-definite matrix defined for each point x in the domain. We impose the Robin boundary condition
where ν_k is the component of the unit outward normal vector ν in the k-th direction. The system to be solved is
as can be shown using an analogue of Green's identity. The coefficients u_i are still found by solving a system of linear equations, but the matrix representing the system is markedly different from that for the ordinary Poisson problem.
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