A Reduced Coupling-Space Multigrid Schur Preconditioner for Fictitious-Domain Interface Problems

Najwa Alshehri

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A Reduced Coupling-Space Multigrid Schur Preconditioner for Fictitious-Domain Interface Problems

Najwa Alshehri

arXiv:2609.40006v1Sep 30, 2026math.NAcs.NA

TL;DR

The FD–DLM discretization creates a large indefinite saddle-point system, and accurately preconditioning its Schur complement at low cost remains a challenge. This paper uses a multigrid approximation with a reduced Woodbury correction in the active multiplier coupling space. The reported experiments show two or three iterations for the multigrid Woodbury preconditioner, while the baseline worsens under refinement.

Problem
The FD–DLM system is a large indefinite saddle-point problem, and an accurate, low-cost Schur complement approximation remains needed.
Source: §1 · pp. 1–2
Method
The method replaces the exact background inverse with a multigrid approximation and applies a reduced Woodbury correction in the active multiplier coupling space.
Source: §3 · p. 4
Results
The baseline worsens under refinement, while the multigrid Woodbury preconditioner requires only two or three iterations.
Source: §4 · p. 5
Takeaways & Limitations
The reduced multigrid correction captures the background response in the active multiplier coupling space and produces behavior close to the exact background solve.
Source: §4 · p. 6
The reported data are not an end-to-end timing study, and a scalable approximation of B−1 remains future work.
Source: §3 · p. 4, §4 · p. 6

Abstract

from arXiv · show

We study a reduced multigrid Schur preconditioner for the fictitious-domain method with distributed Lagrange multipliers applied to elliptic interface problems. The method replaces the costly background solve by a multigrid approximation and treats the resulting correction only in the multiplier coupling space through a Woodbury formula. Numerical experiments show a marked improvement over the baseline preconditioner and consistently rapid convergence under mesh refinement and coefficient contrast.

1 Introduction

Elliptic interface problems require transmission conditions across material interfaces, while the FD–DLM method avoids fitting the background mesh to the interface. The paper asks whether a multigrid approximation can capture the background contribution to the Schur complement in a reduced coupling space.

Elliptic interface problems model fields across subdomains with changing material coefficients and transmission conditions, including heat conduction, diffusion, and deformation.Source: §1 · p. 1
FD–DLM uses meshes independent of the interface, avoiding mesh regeneration, particularly for moving interfaces.Source: §1 · p. 1
The FD–DLM discretization produces a large indefinite saddle-point system that requires effective preconditioning.Source: §1 · p. 1
The central challenge is approximating the Schur complement accurately at low cost, motivating a multigrid approximation of the background contribution in the reduced coupling space.Source: §1 · p. 2

2 Problem Setting and FD–DLM Formulation

The paper formulates an elliptic interface problem on two subdomains separated by an immersed interface, then enforces agreement between background and immersed fields weakly with a distributed multiplier. Its finite-element discretization uses independent meshes and a discontinuous piecewise-constant multiplier.

The problem couples two material subdomains across an internal interface, with continuity of the solution, balanced normal fluxes, and homogeneous Dirichlet data on the outer boundary.Source: §2 · p. 2
FD–DLM extends the background data and field to the full domain and requires the extension to coincide with the immersed field on the immersed subdomain.Source: §2 · p. 2
A distributed Lagrange multiplier imposes the field-matching constraint weakly through the duality between H1(Ω2) and its dual.Source: §2 · pp. 2–3
Independent shape-regular quadrilateral meshes are used for the background and immersed domains, with no matching condition at the interface.Source: §2 · p. 3
The Q1–(Q1 + B)–P0 discretization uses continuous bilinear background functions, bubble-enriched immersed functions, and discontinuous piecewise-constant multipliers.Source: §2 · p. 3
For β2 > β > 0, the cited prior result establishes a unique discrete solution for this stable element.Source: §2 · p. 3

3 Preconditioner

The preconditioner replaces the exact background inverse with a multigrid V-cycle and confines its correction to the active multiplier coupling space. A reduced Woodbury formula applies the inverse, with a dense inverse of size r × r; scalable approximation of B−1 remains future work.

The Q1–(Q1 + B)–P0 choice uses L2(Ω2) coupling and is a demanding conditioning test; a related formulation has cond(A) ≲ h^-4 under uniform refinement.Source: §3 · p. 3
The exact Schur complement requires an inner solve, while preconditioning it by B−1 does not yield mesh-independent behavior under refinement.Source: §3 · p. 4
Replacing the exact background inverse with one multigrid V-cycle yields a Schur approximation whose correction acts only on the coupled multiplier directions.Source: §3 · p. 4
The reduced multigrid correction captures the background response in the active multiplier coupling space, changing only the multiplier–multiplier block.Source: §3 · p. 4
The reduced Woodbury inverse requires only a dense r × r inverse and avoids forming the dense Schur correction or applying the exact background inverse.Source: §3 · p. 4
A scalable approximation of B−1 is left for future work.Source: §3 · p. 4

4 Numerical Results

The multigrid Woodbury preconditioner converges in two or three iterations while the baseline worsens under refinement. The tests also report a compact reduced space, accurate corrections, and limitations on timing interpretation.

4 Numerical Results

The reduced space falls from 18.1% to 11.8% of the full space, while both multigrid-correction errors stay below 0.5%.Source: §4 · p. 5, Table 1 · p. 5
The condition number of the multigrid-preconditioned Schur system remains close to one, even at larger coefficient contrast.Source: §4 · p. 5, Table 2 · p. 5
Two or three GMRES iterations for the multigrid Woodbury preconditioner contrast with a baseline that worsens under refinement.The exact reduced Woodbury inverse takes one iteration on every level.Source: §4 · p. 5, Table 2 · p. 5
On the finest grid, retaining almost all correction energy does not guarantee fast convergence, and the exact-background LU fill ratio grows.Source: §4 · p. 5, Table 3 · p. 6
The reported data are not an end-to-end timing study because multigrid matrices were exported before the MATLAB projection.Source: §4 · p. 6
The reduced multigrid correction captures the background response in the active multiplier coupling space and produces behavior close to that of the exact background solve.Tests used contrasts β2 = 10 and β2 = 100; B−1 was applied exactly in all tests.Source: §4 · pp. 5–6