[Paper Review] Cross-Points in Domain Decomposition Methods with a Finite Element Discretization

Martin J. Gander, Kévin Santugini-Repiquet|arXiv (Cornell University)|Apr 18, 2014

Advanced Numerical Methods in Computational Mathematics15 references16 citations

TL;DR

This paper addresses the challenge of consistently discretizing Neumann conditions at cross-points—where more than two subdomains meet—in non-overlapping domain decomposition methods using finite elements. It proposes two methods: the auxiliary variable method, which introduces additional interface variables for stable convergence, and complete communication, which ensures consistent Neumann exchange by involving all subdomains sharing a cross-point; both enable convergence proofs and optimal parameter selection, with mass lumping shown to yield higher-order Ventcell-like transmission conditions algebraically.

ABSTRACT

Non-overlapping domain decomposition methods necessarily have to exchange Dirichlet and Neumann traces at interfaces in order to be able to converge to the underlying mono-domain solution. Well known such non-overlapping methods are the Dirichlet-Neumann method, the FETI and Neumann-Neumann methods, and optimized Schwarz methods. For all these methods, cross-points in the domain decomposition configuration where more than two subdomains meet do not pose any problem at the continuous level, but care must be taken when the methods are discretized. We show in this paper two possible approaches for the consistent discretization of Neumann conditions at cross-points in a Finite Element setting.

Motivation & Objective

• To resolve the inconsistency in discretizing Neumann conditions at cross-points in finite element domain decomposition methods.
• To provide a mathematically consistent and convergent discretization strategy for Neumann traces at cross-points where multiple subdomains meet.
• To demonstrate convergence of non-overlapping Optimized Schwarz Methods with finite element discretization despite cross-point singularities.
• To show how mass lumping can algebraically generate higher-order Ventcell transmission conditions, avoiding complex tangential derivative discretizations.
• To identify optimal Robin parameters and overlumping factors that minimize convergence oscillation in the presence of cross-points.

Proposed method

• Introduces the auxiliary variable method, where subdomains store additional interface variables to represent Neumann data, enabling consistent Neumann trace discretization at cross-points.
• Employs the complete communication method, requiring information exchange with all subdomains touching a cross-point, even those only connected at a point, to maintain consistency.
• Uses energy estimates to prove convergence of the domain decomposition algorithm when using auxiliary variables, a result not achievable with standard finite element discretizations at cross-points.
• Proposes a method to select among multiple possible Neumann trace splittings at cross-points that minimizes oscillation in the iteration process.
• Demonstrates that mass lumping of Robin conditions yields equivalent higher-order Ventcell transmission conditions algebraically, even at cross-points, avoiding direct discretization of tangential derivatives.
• Performs numerical experiments on Laplace’s equation with $Q_1$ finite elements and $2\times2$ subdomains to evaluate convergence factors under various Robin parameters and overlumping levels.

Experimental results

Research questions

• RQ1 How can Neumann conditions be consistently discretized at cross-points in finite element domain decomposition methods?
• RQ2 Can convergence be proven for non-overlapping Optimized Schwarz Methods with finite element discretization when cross-points are present?
• RQ3 What is the optimal way to split Neumann traces at cross-points to minimize oscillation in the iteration process?
• RQ4 Can higher-order Ventcell transmission conditions be obtained algebraically from Robin conditions via mass lumping in a finite element context?
• RQ5 How do optimal Robin parameters and overlumping factors scale with mesh refinement in the presence of cross-points?

Key findings

• The auxiliary variable method enables a rigorous convergence proof via energy estimates, which is not possible with standard finite element discretizations at cross-points.
• The complete communication method ensures consistent Neumann condition discretization by involving all subdomains sharing a cross-point, even those connected only at a point.
• For the $100\times100$ mesh per subdomain, the best convergence factor $\kappa=0.6013464$ was achieved with $p=2.0$ and $\omega=122.5$ using the complete communication method.
• With the auxiliary variable method, the best convergence factor $\kappa=0.6006753$ was obtained for $100\times100$ cells using $p=2.0$ and $\omega=122.0$.
• The optimal Robin parameter $p$ scales approximately as $O(1/h)$ at cross-points, consistent with previous theoretical estimates, unlike $O(1/\sqrt{h})$ at regular interface points.
• Mass lumping of Robin conditions produces equivalent higher-order Ventcell transmission conditions algebraically, eliminating the need to discretize tangential derivatives, even at cross-points.