This work addresses a novel version of the kernel-free boundary integral (KFBI) method for solving elliptic PDEs with implicitly defined irregular boundaries and interfaces. We focus on boundary value problems and interface problems, which are reformulated into boundary integral equations and solved with the matrix-free GMRES method. In the KFBI method, evaluating boundary and volume integrals only requires solving equivalent but much simpler interface problems in a bounding box, for which fast solvers such as FFTs and geometric multigrid methods are applicable. For the simple interface problem, a correction function is introduced for both the evaluation of right-hand side correction terms and the interpolation of a non-smooth potential function. A mesh-free collocation method is proposed to compute the correction function near the interface. The new method avoids complicated derivation for derivative jumps of the solution and is easy to implement, especially for the fourth-order method in three space dimensions. Various numerical examples are presented, including challenging cases such as high-contrast coefficients, arbitrarily close interfaces and heterogeneous interface problems. The reported numerical results verify that the proposed method is both accurate and efficient.

翻译：本文提出了一种改进版本的无核边界积分方法，用于求解具有隐式定义不规则边界和界面的椭圆型偏微分方程。我们重点研究边值问题和界面问题，将其重新表述为边界积分方程，并采用无矩阵GMRES方法求解。在KFBI方法中，评估边界积分和体积积分仅需在包围盒内求解等效但更简单的界面问题，适用于快速求解器如FFT和几何多重网格方法。针对简单界面问题，引入校正函数用于右端校正项评估和非光滑势函数插值。提出了一种无网格配点方法计算界面附近的校正函数。新方法避免了复杂的解导数跳跃推导，易于实现，尤其适用于三维空间中的四阶方法。展示了多种数值算例，包括高对比度系数、任意接近界面和异质界面问题等具有挑战性的情况。数值结果验证了所提方法的准确性和高效性。