In this paper, a novel local knot method (LKM) is presented to solve the 2D and 3D convection–diffusion–reaction equations in arbitrary domains. Contrary to the traditional boundary knot method, the proposed scheme requires the nodes not only on the boundary but also inside the domain. For each node, we can find a local subdomain containing a certain number of neighboring nodes. Utilizing the non-singular general solution of differential operator and the known boundary conditions, a sparse linear system is established to approximate the solutions at all nodes over the physical domain. The present LKM is a local meshless method with the merits of being mathematically simple, numerically accurate and easy to large-scale computation. Two numerical examples, involving 2D and 3D complicated domains, are provided to illustrate the effectiveness and accuracy of the new methodology.

在本文中，提出了一种新颖的局部结方法（LKM），用于求解任意域中的2D和3D对流-扩散-反应方程。与传统的边界结法相反，提出的方案不仅要求节点在边界上，而且要求在域内。对于每个节点，我们都可以找到包含一定数量的相邻节点的本地子域。利用微分算子的非奇异一般解和已知的边界条件，建立了一个稀疏线性系统来近似物理域上所有节点上的解。当前的LKM是一种局部无网格方法，具有数学上简单，数字精确和易于大规模计算的优点。两个涉及2D和3D复杂域的数值示例，