Ensemble Domain Decomposition Algorithm for the Fully-mixed Random Stokes-Darcy Model with the Beavers-Joseph Interface Conditions

Abstract

In this paper, an efficient ensemble domain decomposition algorithm is proposed for fast solving the fully-mixed random Stokes-Darcy model with the physically realistic Beavers-Joseph (BJ) interface conditions. We utilize the Monte Carlo method for the coupled model with random inputs to derive some deterministic Stokes-Darcy numerical models and use the idea of the ensemble to realize the fast computation of multiple problems. One remarkable feature of the algorithm is that multiple linear systems share a common coefficient matrix in each deterministic numerical model, which significantly reduces the computational cost and achieves comparable accuracy with the traditional methods. Moreover, by domain decomposition, we can decouple the Stokes-Darcy system into two smaller sub-physics problems naturally. Both mesh-dependent and mesh-independent convergence rates of the algorithm are rigorously derived by choosing suitable Robin parameters. Especially, for small hydraulic conductivity in practice, the almost optimal geometric convergence can be obtained by finite element discretization. Finally, two groups of numerical experiments are conducted to validate and illustrate the exclusive features of the proposed algorithm.

Parallel Domain Decomposition method for the fully-mixed Stokes-dual-permeability fluid flow model with Beavers-Joseph interface conditions

“…Through some tests, PDDM seems difficult to converge for δ S ≤ δ D while the intrinsic permeabilities k D and k M are small. Inspired by the numerical experiments in [16,18] and our trying [31], we find that the Robin parameters δ S > δ D might provide decent convergence results. So let δ S = 5.0 and δ D = 1.0 in this simulation to get convergence result.…”

“…This lemma is much general but novel, and such lemma or its variants with more different iterative sequence norms (which could be further proved by mathematical induction) can be applied to other steady-state problems. For example, the similar lemma is developed for the ensemble DDM algorithm of the random Stokes-Darcy model [31]. Such lemma plays a similar role with the Gronwall lemma, which is usually used for the unsteady explicit-implicit decoupled algorithm, but fails for the steady-state system.…”