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 \(C\rho h^{-2}\), where \(\rho \) is the jump of the discontinuous coefficients, h is the mesh size, and the constant C is independent of \(\rho \) and the location of the interface on the triangulation. Numerical results are also given to verify our theoretical findings.

众所周知，像共轭梯度法这样的传统迭代方法的收敛速度取决于刚度矩阵的条件数。此外，诸如多网格和域分解方法之类的快速求解器的构造也需要估计刚度矩阵的条件数。本文的主要目的是对高对比度界面问题的线性和双线性浸入式有限元逼近给出刚度矩阵的严格条件数估计。结果表明，条件数为\（C \ rho h ^ {-2} \），其中\（\ rho \）是不连续系数的跳跃，h是网格大小，常数C与\（\ rho \）和接口在三角剖分上的位置。数值结果也证明了我们的理论发现。