Numerical analysis of the rectangular domain decomposition method

Abstract

SUMMARYWhen solving parabolic partial differential equations using finite difference non-overlapping domain decomposition methods, one often uses the stripwise decomposition of spatial domain and it can be extended to the rectangular decomposition without further analysis. In this paper, we analyze the rectangular decomposition when the modified implicit prediction (MIP) algorithm is used. We show that the performance of the rectangular decomposition and the stripwise decomposition is different. We compare spectral radius, maximum error, efficiency, and total operations of the rectangular and the stripwise decompositions. We investigate the accuracy of the interface of the rectangular decomposition and the effects of the correction phase of the rectangular decomposition. Numerical experiments have been done in both two and three spatial dimensions and show that the rectangular decomposition is not better than the stripwise decomposition.

A Substructuring Domain Decomposition Scheme for Unsteady Problems

“…This numerical implementation (stages 1-3) of regionally-additive scheme (45), (48), (49) is nothing but the scheme of the domain decomposition [31,32,33,34,35,36]) with the explicit-implicit procedure for calculating the boundary conditions at the boundaries of subdomains.…”

“…Among other domain decomposition methods for solving boundary value problems for parabolic equations it is necessary to highlight explicit-implicit methods considered in many papers (see, for example, [31,32,33,34,35,36]).…”

Second‐order treatment of the interface of domain decomposition method for parabolic problems

“…The DD methods developed for elliptic problems include pseudo-spectral method [1], additive/multiplicative Schwarz methods [2,3] and the finite element tearing and interconnecting (FETI) method [4,5]. For parabolic problems, the DD methods include an explicit/implicit Galerkin method [6], a stabilized explicit Lagrange multiplier method [7], Dawson's method [8,9], the explicit prediction and implicit correction (EPIC) method [10], the stabilized explicit/implicit domain decomposition (SEIDD) method [11], the alternating explicit/implicit domain decomposition (AEIDD) method [12], the implicit prediction and implicit correction (IPIC) method [13] and the modified implicit prediction (MIP) method [14,15]. Also, the DD method has been applied to solve hyperbolic problems such as in [16][17][18][19][20][21][22].…”

“…In the spatial domain, the problem is decomposed stripwise [8,10,13,14] or in rectangular manner [15] with overlapping subdomains [23,24] or non-overlapping subdomains [3,8,10,11,13,14].…”

Novel Finite Difference Discretization of Interface Boundary Conditions for Stablized Explicit-Implicit Domain Decomposition Methods

“…Jun and Mai [20] [39] used special treatment for the implicit discretization at points neighboring intersection points while maintaining unconditional stability. The interface boundary treatment introduced by Jun and Mai for their modified implicit prediction method [20] [39] can also be used to solve the intersecting interior boundary problem for corrected EIDD methods. Zhuang and Sun [35] and Wang, Wu, and Zhuang [37] tackled disadvantages of no-crossover interface boundaries by using a data partition different from the domain partition, where the domain is partitioned with no crossover interface boundaries as in Figure 1(b) but the data of each subdoman is further partitioned into multiple data subsets like in Figure 1(a) for distribution to different processors.…”