Triangular surface-based 3D IIM (Immersed Interface Method) algorithms face major challenges due to the need to calculate surface derivative of jumps. This paper proposes a fast, easy-to-implement, surface-derivative-free of jumps, augmented IIM (AIIM) with triangulated surfaces for 3D Helmholtz interface problems for the first time, which combines the simplified AIIM with domain decomposed and embedding techniques. The computational domain is divided into sub-domains along the interface and the solutions of sub-domains are continuously extended into larger regular domains by embedding. The jumps in normal derivative of solution along the interfaces in the extended domains are introduced as unknowns to impose the original jump relations. The original problem is simplified into Helmholtz interface problems with constant coefficients by coupling them with the augmented equation, which is then solved using fast simplified AIIM. This approach eliminates the need to compute surface derivatives of jumps, making implementation of 3D IIM based on triangulated surfaces fairly simple. Numerical results demonstrate that the algorithm is efficient and can achieve the overall second-order accuracy.

中文翻译：

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一种新颖的无跳跃表面导数 AIIM，具有用于 3D 亥姆霍兹界面问题的三角表面

由于需要计算跳跃的表面导数，基于三角形表面的 3D IIM（沉浸式接口方法）算法面临重大挑战。本文首次针对 3D 亥姆霍兹界面问题提出了一种快速、易于实现、表面导数无跳跃的增强型 IIM（AIIM），该方法将简化的 AIIM 与域分解和嵌入技术相结合。将计算域沿界面划分为子域，子域的解通过嵌入不断扩展到更大的规则域。解的法向导数沿着扩展域中的界面的跳跃被引入作为未知数以施加原始跳跃关系。通过将原始问题与增广方程耦合，将其简化为具有常系数的亥姆霍兹界面问题，然后使用快速简化的 AIIM 进行求解。这种方法消除了计算跳跃的表面导数的需要，使得基于三角表面的 3D IIM 的实现相当简单。数值结果表明该算法是高效的并且能够达到整体二阶精度。