An augmented immersed finite element method is proposed for solving elliptic interface problems with discontinuous and variable coefficients. By introducing the jump of normal derivative of the solution along the interface as an augmented variable, the interface problem is solved iteratively on Cartesian meshes using the generalized minimal residual (GMRES) iterative method. Within each iteration, we need to solve a system of linear equations with the same symmetric and positive definite matrix but with different right-hand sides. The sequence of linear equations is solved efficiently by the multi-grid solvers since the set up of coarse grid matrices and decompositions is only computed once. Extensive numerical examples show that the number of GMRES iterations is nearly independent of the mesh size when a diagonal-like preconditioning is applied. Thus the computational work of the proposed method is linearly proportional to the number of grid nodes. Implementation details are given and extensive numerical examples in two and three dimensions are provided to show the accuracy and efficiency of the proposed method.

为了解决具有不连续变系数的椭圆界面问题，提出了一种增强的浸入式有限元方法。通过引入解的正导数沿接口的跃迁作为增变量，使用广义最小残差（GMRES）迭代方法在笛卡尔网格上迭代解决了接口问题。在每次迭代中，我们需要求解一个线性方程组，该线性方程组具有相同的对称和正定矩阵，但右侧不同。线性方程组的序列可通过多网格求解器高效求解，因为粗网格矩阵的建立和分解仅计算一次。大量的数值示例表明，当应用对角线型预处理时，GMRES迭代次数几乎与网格大小无关。因此，所提出方法的计算工作与网格节点的数量成线性比例。给出了实现细节，并提供了二维和三维的大量数值示例，以显示所提出方法的准确性和效率。