A finite element method for singularly perturbed reaction-diffusion problems.
The authors present a stabilized FOLS method for solving the reaction-diffusion equation \(-\kappa\Delta u+\sigma u=f\) for small \(\kappa\) and large \(\sigma\). This equation can be expressed as a system of equations for \(u\) and \(p=-\nabla u\), and then written in terms of the least-squares operator \[ B(u,p;v,q)=\kappa\sigma(p+\nabla u,q+\nabla v)_0+(\kappa\nabla\cdot p+\sigma u,\kappa\nabla\cdot q+\sigma u)_0 \] and a stabilization term \[ T(u,p,v,q)=-\alpha\sum_K h_K^2(\sigma u+\kappa\nabla\cdot p,\sigma v+\kappa\nabla\cdot q)_{0,K}/(\alpha h_K^2+\kappa), \] where the sum is taken over the set of all elements \(K\), each of which has characteristic size \(h_K\). The authors prove that the discretized operator is coercive and that solutions of the discrete system converge. The norm used depends in part on the stabilization term. Furthermore, the condition number of the discrete stiffness matrix is \(O(1/h^2)\), uniformly for \(\sigma\geq\kappa\).
scientific article; zbMATH DE number 1916589