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On the Choice of Continuous and Discontinuous Lagrange Multipliers in Fictitious-Domain Methods

Najwa Alshehri, Shumoua F. Alrzqi

arXiv:2610.03267v1math.NAcs.NA

TL;DR

FD-DLM addresses elliptic interface problems on separate meshes, where complex or dynamic interfaces can make conforming meshes require frequent remeshing. The paper compares two stable low-order formulations using continuous and discontinuous multipliers. Their weak-constraint residuals, mismatch measures, and primal errors differ, so multiplier choice involves an accuracy and system-size trade-off.

  • Problem

    For elliptic interface problems, the study asks which multiplier more strongly enforces discrete coupling, since prior work established stability and solvers but did not resolve that comparison.

    Source: §1 · p. 1

  • Method

    The study compares two stable low-order FD-DLM finite-element triples using continuous and discontinuous multipliers, with bubble enrichment in the discontinuous formulation.

    Source: §2 · p. 2

  • Results

    The discontinuous formulation has near-zero Rmax and lower primal L2 errors, while the continuous formulation typically has smaller local mismatches; at β2 = 100, global mismatch is 4.060 × 10−5 for Element 2 versus 4.296 × 10−5 for Element 1.

    Source: §3 · p. 3, §4 · p. 5

  • Takeaways & Limitations

    Multiplier selection should balance the required accuracy against the size of the resulting system.

    Source: §4 · p. 5

    The findings describe the full stable triples, which differ in multiplier continuity, immersed-primal-space enrichment, and overall system size, not multiplier continuity alone.

    Source: §3 · p. 4

Abstract

from arXiv · show

This article examines the Fictitious-Domain approach using Distributed Lagrange Multipliers (FD-DLM), a method that handles elliptic interface problems with separate meshes without requiring the interface to match the background mesh. It compares two stable, low-order FD-DLM discretizations when using both continuous and discontinuous multiplier spaces. The study distinguishes between satisfying the weak coupling constraint per element and the size of the discrepancy between the two discrete primal fields. Numerical experiments indicate that the continuous formulation usually yields smaller local errors and maximum mismatch values, while the discontinuous formulation shows a more regular decrease in mismatch as the mesh is refined and has smaller primal errors.

1 Introduction

FD-DLM addresses difficult interface discretizations by extending one subdomain to a simple background domain and enforcing agreement with a distributed multiplier. This study asks which multiplier space more strongly enforces discrete coupling, while distinguishing local weak enforcement from mismatch magnitude.

1 Introduction

  • FD-DLM uses separate background and immersed domains, enforcing their agreement with a multiplier distributed over the immersed domain.Extending one subdomain to a simple background domain offers geometric flexibility without sacrificing a variational treatment of interface conditions.Source: §1 · p. 1
  • Prior work established stability and developed solvers for continuous and discontinuous multiplier formulations, but did not determine which more strongly enforces discrete coupling.The cited work includes mixed formulations, a stable discontinuous formulation, an error estimator, and preconditioners [5] [1].Source: §1 · p. 1
  • Elementwise-constant discontinuous multipliers enforce the weak constraint locally, but that property alone does not determine pointwise or root-mean-square mismatch magnitude.The study therefore reports weak-constraint measures, mismatch measures, primal errors, and multiplier degrees of freedom separately from complete system size.Source: §1 · p. 1

2 Interface problem and FD-DLM formulations

The formulations extend a subdomain solution to the background domain and couple the discrete primal fields with a multiplier. They use stable finite-element triples and assess weak-constraint satisfaction separately from mismatch amplitude.

2 Interface problem and FD-DLM formulations

  • FD-DLM extends u1, f1, and β1 from Ω1 across the background domain Ω, with the extension u satisfying u|Ω2 = u2.The setup assumes a bounded Lipschitz domain split into Ω1 and a completely immersed Ω2, with uniformly positive coefficients.Source: §2 · p. 2
  • The discrete multiplier tests the difference between the background-field restriction and the immersed primal field, while the discontinuous space Q0 is elementwise constant.The mismatch is defined as eh := uh|Ω2 − u2,h.Source: §2 · pp. 2–3
  • The FD-DLM system uses two fully stable triples: Element 1, [Q1 − Q1 − Q1], and Element 2, [Q1 − (Q1 + B) − Q0]; Element 2 requires bubble enrichment for stability.The comparison examines fully stable triples rather than the effect of multiplier continuity separately.Source: §2 · p. 2
  • Rmax measures elementwise weak-constraint satisfaction, whereas Emax, E∞max, and Eglobal measure local and global mismatch amplitudes.The reported mismatch metrics use a common set of quadrature points QK.Source: §2 · p. 3

3 Numerical results and interpretation

Across five refinements and both coefficient contrasts, Element 2 has smaller primal L2 errors, while Element 1 often has smaller local mismatch on coarse meshes. The weak-constraint residual and the mismatch measures capture distinct aspects of coupling, and the reported refinement rates describe mismatch decrease rather than primal-error convergence.

3 Numerical results and interpretation

  • The experiment uses an immersed unit disk with β1 = 1 and β2 in {10, 100}.The exact solution is specified for this setup.Source: §3 · p. 3
  • Element 2 has smaller L2 errors in both primal fields across all five refinements and coefficient contrasts; Element 1 tends to have smaller local mismatch on coarse meshes.Table 1(a) reports primal L2 and H1 errors over the refinement levels; Table 1(b) compares residual and mismatch measures on the finest mesh.Source: §3 · p. 3, Table 1 · p. 4
  • Rmax ranges from about 10^-10 to 10^-12 for Element 2, versus values of order 10^-4 to 10^-3 for Element 1.The Element 2 residual matches the cellwise weak identity up to algebraic and quadrature errors.Source: §3 · p. 3
  • Element 2's empirical local-mismatch rates are 0.92–1.08, versus 0.63–0.72 for Element 1; on the finest mesh, local mismatch differs by less than a factor of 1.2.Element 1 tends to have smaller local mismatch on coarse meshes, while Element 2's local mismatch decreases more systematically with refinement.Source: §3 · p. 3
  • For β2 = 100, Eglobal is 4.060 × 10^-5 for Element 2 and 4.296 × 10^-5 for Element 1; empirical global-mismatch rates are 1.53–1.75 and 1.29–1.33, respectively.The rates describe decreases in mismatch measures, not convergence rates for the primal L2 and H1 errors.Source: §3 · p. 3
  • The cellwise weak identity controls element-average mismatch but not its amplitude, because positive and negative contributions can cancel within an immersed element.The mismatch can also be smaller than either individual primal approximation error because those errors may cancel; weak residuals, mismatch measures, and primal errors therefore capture different aspects of coupling.Source: §3 · pp. 3–4

4 Conclusions

The discontinuous formulation yields near-zero Rmax and lower primal L2 errors, while the continuous formulation typically produces smaller local mismatches. Multiplier selection should balance accuracy needs against system size.

4 Conclusions

  • The discontinuous formulation yields near-zero Rmax and lower primal L2 errors, while the continuous formulation typically produces smaller local mismatches.The study concludes that multiplier selection should balance required accuracy with the resulting system size.Source: §4 · p. 5
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