Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains

Abstract

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.