Stiffness matrix

In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix 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 {φ1, ..., φn} defined on Ω which also vanish on the boundary. One then approximates The coefficients u1, ..., un are determined so that the error in the approximation is orthogonal to each basis function φi: The stiffness matrix is the n-element square matrix A defined by By defining the vector F with components Fi = ∫ Ω φ i f d x {displaystyle int _{Omega }varphi _{i}f,dx} , 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. 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.)