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Algebraic convergence analysis for the Interface Control Domain Decomposition (ICDD) method
Marco Discacciati, Paola Gervasio, Alfio Quarteroni
TL;DR
The paper addresses convergence analysis for ICDD applied to elliptic problems with possibly discontinuous coefficients and hp-FEM discretizations. It combines Schur-complement spectral estimates with GMRES theory for conforming overlap discretizations, obtaining mesh-independent theoretical rates and numerical evidence of effectiveness with large coefficient jumps. In this setting, the results also apply to SRAS and RAS.
Problem
Convergence estimates for ICDD, especially their dependence on overlap width, mesh size, polynomial degree, and discontinuous coefficients, are needed for its non-symmetric interface Schur complement system.
Method
The paper analyzes the interface Schur complement through spectral estimates and classical GMRES convergence theory for conforming hp-FEM discretizations on overlapping two-dimensional subdomains.
Results
The theoretical rate is O(δ^-1 p^3/2 log p), independent of h; experiments show p-independent iterations and weaker δ dependence for larger coefficient jumps.
Takeaways & Limitations
In the conforming case, the analysis provides convergence estimates for ICDD and SRAS and, via SRAS equivalence, for RAS.
Takeaways & Limitations
The analysis explicitly treats two overlapping subdomains; dependence on the subdomain size H is not discussed explicitly, and dual ICDD is advantageous only if it at least halves iteration counts because each iteration costs about twice as much.
Abstract
from arXiv · showhide
We develop the convergence analysis of the Interface Control Domain Decomposition (ICDD) method, an overlapping domain decomposition method based on an optimal control framework with Dirichlet interface control functions and interface observation. We consider elliptic problems with possible discontinuous coefficients approximated by $hp-$FEM in each subdomain. When the discretizations are conforming on the overlap between 2D domains, we provide theoretical estimates of the number of GMRES iterations needed to solve the non-symmetric interface Schur complement system associated with ICDD. Our results are obtained by combining novel spectral estimates for the Schur complement matrix of ICDD and classical GMRES convergence theory. We prove that the convergence rate behaves as $\mathcal{O}(δ^{-1} p^{3/2}\log p)$, where $δ$ denotes the overlap width and $p$ the local polynomial degree, while remaining independent of the mesh size $h$. Numerical experiments verify the theoretical predictions and show the effectiveness of ICDD in the presence of large coefficient jumps in the computational domain. Since in the conforming case, the considered ICDD formulation coincides with the Substructured Restricted Additive Schwarz (SRAS) method, the analysis also provides convergence estimates for SRAS in two dimensions and, through its known equivalence, for the Restricted Additive Schwarz (RAS) method.
1 Introduction
The paper analyzes ICDD convergence for elliptic problems with discontinuous coefficients and hp-FEM discretizations, using GMRES for its non-symmetric interface Schur complement. Under conforming overlap discretizations, it derives mesh-independent estimates and extends convergence guarantees to SRAS and RAS.
- Problem and method: ICDD convergence is analyzed for second-order elliptic problems with possibly discontinuous coefficients, Dirichlet interface controls, interface observation, and hp-FEM discretization.The analysis targets dependence on overlap width δ, mesh size h, and local polynomial degree p.
- Algebraic analysis: Because the ICDD Schur complement is non-symmetric, GMRES theory—not condition-number estimates—is used to bound the iterations needed to reach a prescribed residual tolerance.The analysis combines spectral estimates for the interface Schur complement with classical GMRES convergence theory.
- Main convergence result: The theoretical convergence rate is O(δ^-1 p^3/2 log p) and independent of the mesh size h when the two discretizations are conforming on the overlap.Numerical results additionally show iteration counts that are in fact independent of p.
- Coefficient jumps: Large coefficient jumps across the interface weaken the dependence on overlap width δ, with larger jumps associated with fewer iterations.This behavior is reported as a numerical observation.
- Relation to Schwarz methods: In the conforming case, the ICDD formulation coincides with SRAS, so the analysis supplies convergence estimates for SRAS and, through SRAS equivalence, RAS.This addresses prior gaps involving discontinuous coefficients, general discretizations, and two-dimensional domains.
2 The ICDD coupling method
The ICDD method couples overlapping subdomains through Dirichlet interface controls and interface observation, formulated as an optimal control problem. Its formulation includes equivalent systems with or without dual state problems, while harmonic-extension estimates support the convergence analysis and the conforming discretization connects ICDD with SRAS.
- 2.1 Formulation and optimality system: ICDD decomposes the domain into overlapping subdomains whose internal interfaces are separated by overlap width δ, while coefficients may jump across the original subdomain boundary.The local problems impose zero data away from the interfaces and continuity across the interface union.
- 2.1 Formulation and optimality system: Interface functions λ1 and λ2 act as virtual Dirichlet controls for local state problems, and minimizing an interface-observation cost functional recovers the multidomain problem.The controls are traces prescribed on Γ1 and Γ2, and the minimization problem has a unique solution.
- 2.1 Formulation and optimality system: The ICDD control formulation is well posed in a completed trace space rather than necessarily in the original trace space because the cost-functional norm need not make that space complete.This distinction arises because the interface L2 discrepancy is not equivalent to the cost-functional norm on the original trace space.
- 2.1 Formulation and optimality system: The continuous optimality systems with and without dual state problems are equivalent, but their numerical methods differ in stability and computational cost.The dual variant improves stability for very thin overlaps and can be interpreted as a preconditioning step, whereas each dual GMRES iteration costs about twice as much.
- 2.4 Comparison with Schwarz-type methods: For conforming discretizations on the overlap, ICDD coincides with SRAS, extending the convergence analysis to SRAS and, through its known equivalence, to RAS.ICDD exchanges only interface degrees of freedom and avoids assembling the global matrix or sharing global residuals.
3 Spectral properties of the ICDD matrices
The paper reformulates ICDD’s interface equations as a weak Schur complement system, then derives spectral bounds for its matrix and symmetric part under conforming overlap discretizations. The resulting estimates explain the observed dependence on overlap width, mesh size, polynomial degree, and coefficient jumps.
- 3.2 Analysis of the weak Schur complement matrix: The analysis targets GMRES convergence for the non-symmetric ICDD Schur complement by first estimating its spectral properties.The symmetric part is used because the non-symmetric system’s condition number does not directly characterize GMRES convergence.
- 3.2 Analysis of the weak Schur complement matrix: The spectral estimates bound the Schur-complement norm and lower-bound the minimum eigenvalue of its symmetric part using interface mass-matrix estimates independent of h and p.The proof introduces interface mass matrices and derives their spectral equivalence before bounding the Schur-complement quantities.
- 3.2 Analysis of the weak Schur complement matrix: Under conforming overlap discretizations, the weak Schur complement is positive real because its symmetric part is positive definite.Assumption 3 requires matching mesh sizes and polynomial degrees on the overlap.
- 3.3.1 Numerical tests for conforming discretizations: For continuous coefficients, numerical results show ∥eΣ∥2 remains bounded independently of δ, while λmin(eΣs) = O(δ) as δ → 0.Both quantities vary proportionally to hy in the reported mesh-size tests, while the polynomial-degree tests show ∥eΣ∥2 ∼ p^-1.
- 3.3.2 Numerical tests for non-conforming discretizations in the overlap: With discontinuous coefficients, stronger coefficient jumps weaken λmin(eΣs)’s dependence on δ and reduce the resulting dependence of ICDD iterations on overlap width.For discontinuous ν, ∥eΣ∥2 depends mildly on the coefficient ratio, while λmin(eΣs) behaves like p^-2.
4 Convergence of GMRES iterations for the ICDD interface systems
The paper derives GMRES convergence estimates for ICDD Schur complement systems and tests them across conforming, non-conforming, mesh, degree, overlap, and coefficient-jump settings.
- 4 Convergence of GMRES iterations for the ICDD interface systems: GMRES convergence estimates are derived for the weak and strong ICDD Schur complement systems using classical GMRES theory and spectral properties.The analysis targets dependence on overlap width δ, mesh size h, polynomial degree p, and subdomain size.
- 4 Convergence of GMRES iterations for the ICDD interface systems: The weak formulation predicts iteration growth like δ^-1 as δ vanishes, independence from h, and asymptotic growth like p^3/2.The strong-form estimate adds a logarithmic factor in p.
- 4.1 Numerical results: In Test #1, iterations depend on δ more weakly than theory predicts, remain independent of mesh size, and grow slightly less than p^3/2 for weak ICDD.Dual variants use fewer iterations, but their higher per-iteration cost makes them advantageous mainly for very small δ.
- 4.1 Numerical results: For discontinuous coefficients, iterations become less dependent on δ as coefficient jumps increase, while Test #2 shows independence from h and p and Test #3 shows near-independence from γ jumps.The numerical results also report ICDD as the most convenient variant across the considered overlap widths when coefficient jumps increase.
- 4.1 Numerical results: Although outside the convergence theory, non-conforming tests show behavior similar to the conforming case, and the intersecting-interface test yields δ-independent iterations for all ICDD variants.The latter result is reported for the considered Test #5 configurations.
5 Conclusions and future work
The conclusions summarize the spectral and GMRES analysis of ICDD for two overlapping subdomains and identify its applicability to SRAS, while noting scalability limits for many subdomains.
- 5 Conclusions and future work: The paper analyzes ICDD spectral properties and convergence for elliptic problems with possibly discontinuous coefficients in two-dimensional, two-subdomain overlapping decompositions.ICDD is formulated as optimal control minimizing solution jumps on internal interfaces, with a Schur complement whose unknowns are interface quantities.
- 5 Conclusions and future work: For conforming overlap discretizations, the ICDD optimality system coincides with SRAS, so the estimates also apply to SRAS.The paper therefore supplies convergence analysis for SRAS in this setting.
- 5 Conclusions and future work: For many subdomains, the dependence on subdomain size H is represented by cLH, which behaves like H^-1 as H tends to zero.A coarse level is identified as a future extension to remove this dependence, paralleling classical Schwarz methods.
A.1 Equivalence between SRAS and ICDD for conforming meshes
For two conforming overlapping subdomains, the appendix verifies that the ICDD Schur complement system matches the SRAS system, including its matrix, right-hand side, and interface unknowns.
- A.1 Equivalence between SRAS and ICDD for conforming meshes: The appendix partitions degrees of freedom into non-overlap, interface, and overlap blocks for the two conforming subdomains.The notation distinguishes the two interfaces and the shared overlapping region.
- A.1 Equivalence between SRAS and ICDD for conforming meshes: With suitable restriction and prolongation operators, the ICDD system matrix becomes the SRAS Schur complement matrix Σ.The appendix explicitly identifies the resulting matrix with the matrix defined for SRAS.
- A.1 Equivalence between SRAS and ICDD for conforming meshes: The right-hand sides of the two systems also coincide, and the ICDD interface unknown uS coincides with the SRAS variable λ.Together these identifications establish equivalence of the formulations, not merely similarity of their matrices.
A.2 Proof of Theorem 1
The appendix proves the weighted trace and extension estimates needed for the main analysis by combining domain inequalities, coefficient bounds, harmonic extensions, and scaling arguments.
- A.2 Proof of Theorem 1: For harmonic extensions uλ, Green’s formula on Ds and Fubini-based derivative manipulations produce the estimates required by Theorem 1.The extension satisfies the homogeneous elliptic equation in the subdomain and prescribed interface data.
- A.2 Proof of Theorem 1: The proof introduces weighted Sobolev norms and bounds their constants using coefficient extrema and domain geometry.The constants depend on ν, γ, and geometric parameters, with separate treatment when γ is zero.
- A.2 Proof of Theorem 1: The proof concludes by combining the intermediate inequalities and scaling arguments, with the resulting bracketed factor positive and below one.This establishes the stated theorem estimates under the specified assumptions.
- A.2 Proof of Theorem 1: Trace estimates on Dt are obtained from the L2-trace inequality, the domain diameter Ht, and the weighted norm bounds.The argument sets Dt=(0,t)×(0,Hy) and uses the definition of the weighting function.
A.3 Discrete norm
This section introduces Legendre–Gauss–Lobatto quadrature and shows that its discrete norm is equivalent to the continuous L2 norm for degree-p polynomials, with a diagonal mass matrix.
- Legendre–Gauss–Lobatto quadrature uses nodes and weights on [-1,1] to define the discrete inner product.
- The discrete LGL norm is equivalent to the L2 norm for every polynomial of degree at most p.
- The Lagrange basis at the quadrature nodes makes the discrete mass matrix diagonal, with entries given by the quadrature weights.
A.4 Estimate of λmin(eΣs) for non-conforming meshes
For non-conforming meshes, interpolation and stability estimates are developed for traces across the interface, but a positive lower bound for the minimum eigenvalue cannot be guaranteed.
- Non-conforming interface discretizations admit an interpolation estimate with constants independent of local mesh sizes and polynomial degrees.
- A stability estimate is established for the interface interpolation operator under the non-conforming discretization assumptions.
- The trace of a subdomain function on the opposing interface is globally continuous and piecewise polynomial, enabling the interpolation estimates to be applied.
- For non-conforming meshes, no positive lower bound for ζT eΣ ζ is available, so positivity of λmin(eΣs) cannot be ensured.The paper reports numerical confirmation of this behavior in Test #4 of Sect. 3.3.2.
A.5 Convergence theory for GMRES
This section applies a classical GMRES convergence theorem to positive, generally non-symmetric matrices through their symmetric parts, yielding residual bounds and a tolerance-based stopping estimate.
- The GMRES analysis considers a positive real matrix and controls convergence using the symmetric part of that matrix.
- The theorem bounds the relative GMRES residual after m iterations using a parameter β and the spectral properties of the symmetric part.
- The resulting stopping criterion provides an iteration condition for reaching a prescribed tolerance ϵ.
A.6 Practical implementation of ICDD methods
The appendix gives matrix-free implementation procedures for ICDD and dual ICDD, including local assembly, Schur-complement products, right-hand sides, and Krylov iterations.
- The implementation framework supports matrix-free GMRES iterations for both the ICDD and dual ICDD methods.
- Algorithm 1 assembles local stiffness matrices, right-hand-side vectors, and intergrid matrices for the two subdomains.
- The intergrid matrices map subdomain quantities to opposing interfaces and reduce to the conforming formulas when overlap meshes match.
- The remaining procedures assemble the Schur-complement right-hand side, perform matrix-vector products, and organize the Krylov method.
- Algorithms 6–8 provide the corresponding implementation procedures for the dual ICDD method.