The Stiffness Matrix For Other Problems
Determining the stiffness matrix for other PDE 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 A(x) = akl(x) is a positive-definite matrix defined for each point x in the domain. We impose the Robin boundary condition
where νk 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 ui 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.
In general, to each scalar elliptic operator L of order 2k, there is associated a bilinear form B on the Sobolev space Hk, so that the weak formulation of the equation Lu = f is
for all functions v in Hk. Then the stiffness matrix for this problem is
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