In this paper, a novel collocation method is presented for the efficient and accurate evaluation of the two-dimensional elliptic partial differential equation. In the new method, the physical domain is discretized into a series of overlapping small (local) subdomains, and in each of the subdomain, a localized Chebyshev collocation method is applied in which the unknown functions at every node can be computed by using a linear combination of unknowns at its near-by nodes. The Chebyshev polynomials employed here can provide the spectral accuracy of new approach. The concept of the local subdomain is introduced to derive a sparse system, which ensures the feasibility for large-scale simulation. This paper aims at proposing a new method to solve general partial differential equations accurately and efficiently. Several numerical examples including Poisson equation, Helmholtz-type equation and transient heat conduction equation are provided to demonstrate the validity and applicability of the proposed method. Numerical experiments indicate that the localized Chebyshev collocation method is very promising for the efficient and accurate solution of large-scale problems.

本文提出了一种新颖的配置方法，用于高效，准确地评估二维椭圆偏微分方程。在新方法中，将物理域离散为一系列重叠的小（局部）子域，并且在每个子域中，都应用了局部Chebyshev配置方法，其中可以使用线性算法来计算每个节点处的未知函数临近节点的未知数的组合。这里采用的切比雪夫多项式可以提供新方法的光谱精度。引入局部子域的概念以导出稀疏系统，从而确保了大规模仿真的可行性。本文旨在提出一种准确有效地求解一般偏微分方程的新方法。给出了包括泊松方程，亥姆霍兹型方程和瞬态热传导方程在内的几个数值例子，证明了该方法的有效性和适用性。数值实验表明，局部切比雪夫搭配方法对于有效，准确地解决大规模问题是非常有前途的。