Local knot method for 2D and 3D convection-diffusion-reaction equations in arbitrary domains

Abstract 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.