This paper presents a Cartesian grid based tailored finite point method (TFPM) for singularly perturbed reaction-diffusion equation on complex domains. The method is incorporated with the kernel-free boundary integral algorithm, where the semi-discrete boundary value problems after time integration are reformulated into corresponding Fredholm boundary integral equations (BIEs) of the second kind, however with no algorithmic dependence on the exact analytical expression for the kernels of integrals. The BIEs are iteratively solved by the GMRES method while integral evaluation during each iteration resorts to solving an equivalent interface problem, which in practice is achieved by a series of manipulations in the framework of TFPM including discretization, correction, solution, and interpolation. The proposed method has second-order accuracy for the reaction-diffusion equation as demonstrated by the numerical examples.

本文提出了一种基于笛卡尔网格的定制有限点方法 (TFPM)，用于复杂域上的奇异扰动反应扩散方程。该方法与无核边界积分算法相结合，其中时间积分后的半离散边界值问题被重新表述为相应的第二类 Fredholm 边界积分方程 (BIE)，但算法不依赖于精确的解析表达式对于积分的核。BIEs 由 GMRES 方法迭代求解，而每次迭代中的积分评估则求助于求解等效接口问题，这实际上是通过在 TFPM 框架下的一系列操作来实现的，包括离散化、校正、求解和插值。