2024/08/29 by Meng, Yanzun, Shi, Zuoqiang
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2408.16243
This paper presents an asymptotically compatible error bound for the finite element method (FEM) applied to a nonlocal diffusion model. The analysis covers two scenarios: meshes with and without shape regularity. For shape-regular meshes, the error is bounded by \(O(hk + δ)\), where \(h\) is the mesh size, \(δ\) is the nonlocal horizon, and \(k\) is the order of the FEM basis. Without shape regularity, the bound becomes \(O(hk+1/δ+ δ)\). In addition, we present an efficient implementation of the finite element method of nonlocal model. The direct implementation of the finite element method of nonlocal model requires computation of 2n-dimensional integrals which are very expensive. For the nonlocal model with Gaussian kernel function, we can decouple the 2n-dimensional integral to 2-dimensional integrals which reduce the computational cost tremendously. Numerical experiments verify the theoretical results and demonstrate the outstanding performance of the proposed numerical approach.