A Posteriori Stopping Criteria for Optimized Schwarz Domain Decomposition Algorithms in Mixed Formulations

Abstract

This paper develops a posteriori estimates for domain decomposition methods with optimized Robin transmission conditions on the interface between subdomains. We choose to demonstrate the methodology for mixed formulations, with a lowest-order Raviart–Thomas–Nédélec discretization, often used for heterogeneous and anisotropic porous media diffusion problems. Our estimators allow to distinguish the spatial discretization and the domain decomposition error components. We propose an adaptive domain decomposition algorithm wherein the iterations are stopped when the domain decomposition error does not affect significantly the overall error. Two main goals are thus achieved. First, a guaranteed bound on the overall error is obtained at each step of the domain decomposition algorithm. Second, important savings in terms of the number of domain decomposition iterations can be realized. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive stopping criteria.

A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types

“…Second, we consider ϕ −1 i (I av (ϕ k,n,m h,i )(x)) at the Lagrange degrees of freedom x to define s k,n,m h ; we modify it at the interface Γ, on the boundaries Γ D i , and in the element interiors to satisfy is not H(div, Ω)-conforming (it does not lie in the space RTN 0 (Ω)), and cannot be used as a flux reconstruction in the sense of (5.3). To obtain σ k,m hτ , the procedure used in the linear case in [6,7] can be employed here. The details are given in Appendix A.…”

“…We provide here the details on the reconstruction of an equilibrated flux satisfying (5.3), following closely [6,7]. For each subdomain Ω i , we consider a subset B i of Ω i , termed a band, which contains all the elements of T h,i that share a face, and edge, on a point with the interface Γ.…”

“…In practice, this means that the a posteriori error estimates we develop have to hold true at each iteration of the OSWR algorithm and the linearization, and distinguish the different error components (space, time, linearization, and domain decomposition). For this part of our work, we take up the path initiated in [9,11,26,38,54] for general techniques taking into account inexact algebraic solvers, [45] for (multiscale) mortar techniques, [46,47] for FETI and BDD domain decomposition algorithms combined with conforming finite element discretizations, and most closely [6,7] for respectively steady and unsteady linear problems with similar space and time discretizations. To achieve our goals here, we use a) equilibrated H(div)-conforming flux reconstructions, piecewise constant in time, that are direct extension of [7,25,54] to our model; b) novel subdomain-wise H 1 -conforming saturation reconstructions, continuous and piecewise affine in time, which are targeted to the present setting in that they satisfy the nonlinear and discontinuous interface conditions.…”

A posteriori stopping criteria for space-time domain decomposition for the heat equation in mixed formulations

“…In the context of mixed finite elements, which are mass conservative and which can handle well heterogeneous and anisotropic diffusion tensors, we refer also to [22,38,40]. The multi-domain problem can actually be reformulated as an interface problem (see [20], [38], or [3]) that can be solved by various iterative methods such as the block-Jacobi or GMRES method.…”

“…For domain decomposition strategies with more general interface conditions and where neither the conformity of the flux nor that of the potential is preserved (as long as the convergence is not reached), a new adaptive domain decomposition algorithm has been introduced in [3]. More precisely, three reconstructions are proposed: a flux reconstruction that is globally H(div, Ω)-conforming and locally conservative in each mesh element based on the construction of [54,Section 3.5.2] as well as two H 1 -conforming potential reconstructions, one globally on Ω relying on the averaging operator I av (see [1,13,43]) and another on each subdomain Ω i , which introduces weights on the interfaces and whose goal is to separate the DD and the discretization components.…”