2-Dimensional primal domain decomposition theory in detail

Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions

“…Since it is an FEM discretization of an elliptic problem, we can employ a spatial DDM to increase the parallelism. We opt for a Schur complement method [1], [14] in the 2-dimensional case. Since in the 1-dimensional counterpart the Schur complement method is simply the direct solver, we employ an overlapping Schwarz method to verify the robustness of the parareal.…”

“…We shall describe the domain decomposition method referring to Figure 1. We follow the presentation in [14]. We assume that besides the FEM triangulation, the computational domain Ω ⊂ R 2 is decomposed into N nonoverlapping triangular or rectangular subdomains Ω i , i = 1, .…”

“…The idea of the DDM of our choice, the so-called vertex-based method [1], [23], [14], is to replace the costly Schur complement S by an approximation S. We rely on the observation that S is a blockwise sparse matrix with the sparsity pattern corresponding to the graph of the skeleton. Namely, only the pairs of basis functions that are associated to a common subdomain Ω i have a nonzero contribution in S.…”

“…Further, the action of S −1 involves the solution of a coarse-grid problem arising from the same operator, which is now discretized by the FEM on the DDM skeleton grid. It is proven in [14] that the condition number of this preconditioned system is…”