Approximate Error Bounds on Solutions of Nonlinearly Preconditioned PDEs

Abstract

In many multiphysics applications, an ultimate quantity of interest can be written as a linear functional of the solution to the discretized governing nonlinear partial differential equations and finding a sufficiently accurate pointwise solution may be regarded as a step toward that end. In this paper, we derive a posteriori approximate error bounds for linear functionals corresponding to quantities of interest using two kinds of nonlinear preconditioning techniques. Nonlinear preconditioning, such as the inexact Newton with backtracking and nonlinear elimination algorithm and the multiplicative Schwarz preconditioned inexact Newton algorithm, may be effective in improving global convergence for Newton's method. It may prevent stagnation of the nonlinear residual norm and reduce the number of solutions of large ill-conditioned linear systems involving a global Jacobian required at each nonlinear iteration. We illustrate the effectiveness of the new bounds using canonical nonlinear PDE models: a flame sheet model and a nonlinear coupled lid-driven cavity problem.

A Nonlinear Elimination Preconditioned Inexact Newton Algorithm

“…= \emptyse and S (0) g = S, and define bad components as unknowns whose local Mach numbers exceed a given value, here M j > 0.45, an evolving contiguous range around the shock as it develops. The Jacobian systems for both global and subspace problems are solved by GMRES (30) with RAS preconditioning of overlap 2. The termination tolerances are set as \epsilon global - nonlinear - rtol = 10 - 10 , \epsilon global - linear - rtol = 10 - 3 , \epsilon sub - nonlinear - rtol = 10 - 2 , and \epsilon sub - linear - rtol = 10 - 3 .…”

“…Both global systems and subspace nonlinear problems are solved by INB techniques. Global Jacobian systems are solved by GMRES (30) with right overlapping restricted additive Schwarz preconditioners, where each individual block is solved by the direct LU decomposition and the overlap is set to 2. Our tests set the initial guess to be a simple interpolation of the farfield boundary condition, i.e., \Phi (x, y) = x, for both INB and NEPIN.…”

“…Left nonlinear preconditioners lead to a system with the same root as the original system, which is solved by an outer Jacobian-free Newton method [26]. Examples include the ad-ditive (and multiplicative) Schwarz preconditioned inexact Newton methods ASPIN (MSPIN) [1,3,5,22,38] and two-level ASPIN [6,34] and MSPIN [28,29,30], and the restricted nonlinear Schwarz preconditioners RASPEN [12,16] and SRASPEN [9]. As with linear preconditioning, a left-preconditioned Jacobian is generally not formed explicitly, it being typically much denser than the original; only the matrix-vector multiplication is provided for Krylov subspace methods.…”

“…On the other hand, right nonlinear preconditioners are often associated with a nonlinear elimination (NE) [27] procedure, such as that described in [10] and in nonlinearly preconditioned inexact Newton methods [14,20,30,40], which have been applied effectively to such challenging problems as incompressible Navier--Stokes equations at high Reynolds numbers [32], blood flow in branching arteries [33], and two-phase flow in porous media [41]. Many variants also attract increasing attention, such as nonlinear FETI (finite element tearing and interconnecting) [35], nonlinear FETI-DP (FETI-dual primal) [23,24,25] and nonlinear BDDC (balancing domain decomposition by constraints) [23,25].…”