A Rigorous Condition Number Estimate of an Immersed Finite Element Method - Wang, Saihua; Wang, Feng; Xu, Xuejun (Journal of scientific computing, 2020)
It is known that the convergence rate of the traditional iteration methods like the conjugate gradient method depends on the condition number of the stiffness matrix. Moreover the construction of fast solvers like multigrid and domain decomposition methods also need to estimate the condition number of the stiffness matrix. The main purpose of this paper is to give a rigorous condition number estimate of the stiffness matrix resulting from the linear and bilinear immersed finite element approximations of the high-contrast interface problem. It is shown that the condition number is C ρ h - 2 , where ρ is the jump of the discontinuous coefficients, h is the mesh size, and the constant C is independent of ρ and the location of the interface on the triangulation. Numerical results are also given to verify our theoretical findings.