[Paper Review] Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems

Marcella Bonazzoli, Xavier Claeys+2 more|arXiv (Cornell University)|Mar 20, 2020

Matrix Theory and Algorithms29 references1 citations

TL;DR

This paper presents a rigorous convergence analysis of the SORAS (Symmetrized Optimized Restricted Additive Schwarz) domain decomposition preconditioner for non-self-adjoint or indefinite linear systems, generalizing existing theory beyond symmetric positive definite problems. By establishing bounds on the norm of the preconditioned matrix and the distance of its field of values from the origin, the authors prove convergence for GMRES applied to the reaction-convection-diffusion equation with Robin-type transmission conditions, without requiring a coarse mesh or restrictive assumptions on coefficients.

ABSTRACT

We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).

Motivation & Objective

• To develop a general convergence framework for non-self-adjoint or indefinite problems where standard spectral analysis fails.
• To extend the field of values-based analysis of domain decomposition preconditioners beyond symmetric positive definite problems.
• To establish convergence bounds for the SORAS preconditioner applied to the heterogeneous reaction-convection-diffusion equation with Robin-type transmission conditions.
• To prove robustness of the method under varying convection, reaction, and diffusion coefficients without requiring a coarse mesh.

Proposed method

• Generalizing the field of values approach from the Helmholtz equation to a broad class of non-self-adjoint and indefinite problems.
• Defining a set of assumptions and estimates sufficient to bound the norm of the preconditioned matrix and the distance of its field of values from the origin.
• Applying the framework to the reaction-convection-diffusion equation with a coercive, non-symmetric bilinear form.
• Using Robin-type transmission conditions on subdomain interfaces to construct the SORAS preconditioner.
• Proving stability and convergence under minimal assumptions on physical coefficients and numerical parameters.
• Validating the theory numerically with FreeFEM and the ffddm framework, including SUPG stabilization for convection-dominated regimes.

Experimental results

Research questions

• RQ1 Can the field of values framework be generalized to provide convergence bounds for non-self-adjoint and indefinite problems?
• RQ2 What conditions ensure that the SORAS preconditioner yields a bounded preconditioned matrix and a field of values bounded away from the origin?
• RQ3 Can convergence bounds be rigorously established for the SORAS preconditioner applied to the reaction-convection-diffusion equation with Robin transmission conditions?
• RQ4 Is the convergence robust with respect to convection strength, reaction coefficient, and viscosity, especially in convection-dominated regimes?
• RQ5 Does the method remain effective without relying on a coarse mesh with small elements?

Key findings

• The SORAS preconditioner achieves convergence for the reaction-convection-diffusion equation with Robin-type transmission conditions, even when the bilinear form is non-symmetric and coercive.
• The number of GMRES iterations remains bounded and robust under varying convection and reaction coefficients, with ORAS generally outperforming SORAS in numerical tests.
• For convection-dominated problems with a = [1, 0]T, the standard Galerkin method exhibits instabilities, but SUPG stabilization restores robustness and enables convergence.
• Weak scaling tests show that iteration counts grow with the number of subdomains N, especially for small viscosity ν = 0.001, indicating the need for two-level methods.
• The convergence of SORAS and ORAS is improved with larger overlap δ, and both methods remain effective even when the stability condition on ˜c = c₀ − 1 is violated.
• The theoretical framework does not require a coarse mesh with small elements, distinguishing it from prior two-level approaches.