We study domain decomposition counterparts of the classical alternating direction implicit (ADI) and fractional step (FS) methods for solving the large linear systems arising from the implicit time stepping of parabolic equations. In the classical ADI and FS methods for parabolic equations, the elliptic operator is split along coordinate axes; they yield tridiagonal linear systems whenever a uniform grid is used and when mixed derivative terms are not present in the differential equation. Unlike coordinate-axes-based splittings, we employ domain decomposition splittings based on a partition of unity. Such splittings are applicable to problems on nonuniform meshes and even when mixed derivative terms are present in the differential equation and they require the solution of one problem on each subdomain per time step, without iteration. However, the truncation error in our proposed method deteriorates with smaller overlap amongst the subdomains unless a smaller time step is chosen. Estimates are presented for the asymptotic truncation error, along with computational results comparing the standard Crank-Nicolson method with the proposed method.
SIAM Journal on Scientific Computing
Mathew, T. P.; Polyakov, Peter; Russo, G.; and Wang, J. (1998). "Domain Decomposition Operator Splittings for the Solution of Parabolic Equations." SIAM Journal on Scientific Computing 19.3, 912-932.