Multilevel BDDC for Incompressible Navier--Stokes Equations

Martin Hanek,Pavel Burda,Jakub Šístek

doi:10.1137/19m1276479

Abstract

We present an approach to the numerical solution of steady Navier--Stokes equations. Approximation by the finite element method (FEM) leads to a nonlinear saddle-point system. The system is linearized by the Picard iteration, which leads to a sequence of linear saddle-point systems with nonsymmetric matrices. In this paper, we study the application of Balancing Domain Decomposition based on Constraints (BDDC) to these systems. In particular, we formulate the multilevel BDDC method and explore its applicability for the benchmark problem of lid-driven cavity. Another contribution of the paper is describing the development and application of our BDDC solver to real-world problems of oil flow in hydrostatic bearings.

Highlights

The Balancing Domain Decomposition based on Constraints (BDDC) was introduced in [8] for the Poisson problem and linear elasticity
The multilevel BDDC method was combined with the adaptive selection of coarse unknowns and implemented into an open-source parallel solver BDDCML in [26]
Since we are mainly interested in the efficiency of the BDDC method and the linear solver, we focus on the mean number of linear iterations over all nonlinear iterations, the mean setup time for preparing the BDDC preconditioner, the mean time for solving the linear problem, and the mean time for one linear iteration

Summary

Introduction

The Balancing Domain Decomposition based on Constraints (BDDC) was introduced in [8] for the Poisson problem and linear elasticity. In the 2-level method, we solve (4.8) by a direct method, whereas in the multilevel method, we apply a step of the BDDC method to solve the coarse problem (4.8) only approximately This means that we build residuals and matrices (in setup) corresponding to the cluster of subdomains, which will form one subdomain on the level. We compare the behavior of the 2level BDDC method for different types of interface weights described, namely the cardinality scaling (card ), scaling by diagonal stiffness (diag), scaling weights from unit load (ul ), and the proposed upwind weights. For these simulations, the number of subdomains is 125 with eight elements per subdomain edge. Recall that there is no sliding considered in this case

Conclusions

Findings

Methods