Anisotropic diffusion is important to many different types of common materials and media. Based on structured Cartesian meshes, we develop a three‐dimensional (3D) nonhomogeneous immersed finite‐element (IFE) method for the interface problem of anisotropic diffusion, which is characterized by an anisotropic elliptic equation with discontinuous tensor coefficient and nonhomogeneous flux jump. We first construct the 3D linear IFE space for the anisotropic nonhomogeneous jump conditions. Then we present the IFE Galerkin method for the anisotropic elliptic equation. Since this method can efficiently solve interface problems on structured Cartesian meshes, it provides a promising tool to solve the physical models with complex geometries of different materials, hence can serve as an efficient field solver in a simulation on Cartesian meshes for related problems, such as the particle‐in‐cell simulation. Numerical examples are provided to demonstrate the features of the proposed method.

中文翻译：

---

具有非均匀磁通跳跃的各向异性静磁/静电界面问题的三维浸入有限元方法

各向异性扩散对许多不同类型的常用材料和介质很重要。基于结构笛卡尔网格，我们针对各向异性扩散的界面问题开发了一种三维（3D）非均匀沉浸有限元（IFE）方法，其特征在于具有不连续张量系数和非均匀通量跳变的各向异性椭圆方程。我们首先为各向异性非均匀跳跃条件构造3D线性IFE空间。然后，我们提出了各向异性椭圆方程的IFE Galerkin方法。由于此方法可以有效地解决结构化笛卡尔网格上的界面问题，因此它为解决具有不同材料的复杂几何形状的物理模型提供了一种有前途的工具，因此可以在笛卡尔网格的仿真中作为有效的场求解器，以解决相关问题，例如粒子内仿真。数值例子说明了该方法的特点。