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The geometric convergence of a parallel domain decomposition method on manifolds

Lizhen Qin

arXiv:2609.09911v1math.NAcs.NAmath.DG

TL;DR

Whether the continuous parallel domain decomposition method converges geometrically on manifolds had remained open. This paper proves geometric convergence using a partition of unity and the maximum principle, and derives rate estimates that reflect domain geometry and equation coefficients.

  • Problem

    Whether the continuous parallel DDM converges geometrically on manifolds remained open, despite known convergence results for Euclidean domains and manifolds.

    Source: §1 · p. 2

  • Method

    The proof combines a partition of unity with the classical maximum principle, with an inductive winding-number argument for special cases.

    Source: §1 · p. 2

  • Results

    The general theorem establishes geometric convergence in C0-norm under Assumptions 2.1 and 2.3, and the paper derives rate estimates reflecting domain geometry and the equation.

    Source: §9 · p. 34

  • Takeaways & Limitations

    A continuous DDM with a satisfactory convergence theory can be implemented using effective numerical approaches.

    Source: §1 · p. 3

    With low regularity, smoothness of the error is not guaranteed, and derivative convergence senses and maximal order depend on equation, weight, and boundary regularities.

    Source: §8.3 · p. 34

Abstract

from arXiv · show

This paper establishes a convergence theory for a continuous domain decomposition method for elliptic equations on manifolds. This method originated in \cite{lions2} in the setting of Euclidean domains and was later adapted and generalized to manifolds by \cite{qin_wang_wang}. Although its convergence was known, whether the convergence is geometric has remained open. We prove its geometric convergence and provide various estimates on the convergence rate.

1. Introduction.

The paper analyzes a continuous parallel domain decomposition method for elliptic equations on manifolds, addressing an open question about whether its convergence is geometric. It proves geometric convergence and derives rate estimates, with a proof using a partition of unity and the maximum principle.

1. Introduction.

  • Local coordinate charts let manifold elliptic problems be decomposed into Euclidean subproblems, and numerical DDM implementations have been tested on manifolds of dimensions 4 to 10.Those tests include manifolds with and without boundary, including complicated manifolds that are not submanifolds of Euclidean spaces.Source: §1 · pp. 1–2
  • Geometric convergence of the continuous parallel DDM on manifolds had remained open, although convergence was known for Euclidean domains and had been extended to manifolds.The continuous sequential DDM already had a geometric convergence theory on manifolds.Source: §1 · p. 2
  • The paper proves geometric convergence for the continuous parallel DDM by combining a partition of unity with the classical maximum principle.The method’s partition of unity distinguishes it from Lions’ original method, even on Euclidean domains.Source: §1 · p. 2
  • For special cases of the general C0-convergence theorem, the proof uses a winding number to limit induction along a sequence of subdomains.The argument follows ideas from and traces back to work of Lions.Source: §1 · p. 2
  • The convergence-rate estimates show how domain-decomposition geometry and equation coefficients influence convergence, while C0-convergence implies high-order derivative convergence under relevant regularity assumptions.The results are stated in Theorems 2.6, 2.17, 2.20, 2.22, and 2.24.Source: §1 · p. 2
  • A continuous DDM and convergence theory provide a foundation for implementation with effective numerical approaches, complementing the more prevalent numerical-level DDM literature.Surface problems also have a mature finite-element approach based on global grids.Source: §1 · pp. 1, 3

2. Main Results.

This section formulates the manifold elliptic problem and domain decomposition iteration, then establishes geometric convergence and estimates its rate. It also examines the effects of winding number, coefficients, and decompositions on the rate, and gives regularity results for the iterates under stated assumptions.

2. Main Results.

  • The section formulates the manifold elliptic problem and presents Theorems 2.6, 2.11, 2.14, 2.15, 2.17, 2.20, 2.22, and 2.24 as its main convergence results.Source: §2 · p. 3

2.1. Elliptic Problem.

  • The results apply to the full class of second-order linear elliptic equations on manifolds, since each such operator can be reduced to the stated form using a suitable Riemannian metric.The problem has a unique distributional solution under Assumption 2.1.Source: §2.1 · p. 4

2.2. Domain Decomposition Method.

  • The partition-of-unity iteration is continuous and permits a proof of geometric convergence, unlike the earlier approach whose convergence was not established as geometric.The method was proposed at the continuous level in [41, Algorithm 2.1] and had been numerically validated in and [30].Source: §2 · p. 3, §2.2 · p. 6

2.3. Convergence Rate: Upper Bound.

  • The constant Θ is an upper bound on the iteration’s convergence rate, satisfies 0 < Θ ≤ 1, and is strictly less than 1 if c > 0 somewhere in each subdomain with ∂Mi = γi.Source: §2.3 · p. 6

2.4. Winding Number.

  • The winding number N of the decomposition is the first stabilization index of the iterated subdomain sets, and the sets have reached all subdomains by that index when ∂M ≠ ∅.The winding number reflects decomposition complexity, which affects the convergence rate.Source: §2.4 · p. 7

2.5. C0-Convergence.

  • ∥u_n − u∥_C0(M) ≤ Θ^n∥u_0 − u∥_C0(M), with Θ ≤ 1; if c > 0 somewhere in each specified subdomain, Θ < 1 and convergence is geometric.When ∂M ≠ ∅, the winding number N also gives a geometric estimate over N iterations, with constants independent of n, u_0, and u.Source: §2.3 · p. 6, §2.5 · p. 8
  • For ∂M ≠ ∅, N > 2, and M ≠ S, equation (2.10) gives the C0 error norm as its initial value, ∥u0 − u∥C0(M), for every n < N − 1.Source: §2.5 · p. 8
  • Algorithm 2.1 converges geometrically in full generality, including cases where the earlier convergence theorems do not apply.The general result is covered by Theorem 8.3; the paper gives a brief exposition of its proof idea in §8.1 rather than a full proof.Source: §2.5 · p. 9

2.6. Convergence Rate: Comparison.

  • Increasing c lowers the rate bound Θ, strictly when it increases in every subdomain, and Θ tends to zero as constant c tends to infinity.The non-strict comparison holds when the new coefficient is everywhere at least c.Source: §2.6 · p. 9
  • Larger subdomains can lower Θ, while refinement raises it; strict improvement or deterioration requires the theorem’s stated conditions.For strict improvement under a larger decomposition, Θ(c, D, ρ) must be below 1 and each enlarged subdomain outside its specified boundary portion must contain the original subdomain. Strict refinement raises Θ when the original bound is below 1; a single subdomain yields the solution in one iteration.Source: §2.6 · p. 9

2.7. High Regularity.

  • With ρ_i in Ck,α(M), each iterate error u_n − u lies in Ck,α(M), with an estimate uniform in n, even without assuming u has high regularity.If u itself is in Ck,α(M), then each iterate u_n is also in Ck,α(M).Source: §2.7 · p. 10
  • Theorem 2.24 gives estimates for iterates after n > 1 with Lipschitz partition weights, and a second estimate when a subdomain boundary is Ck,α and the weights are Ck,α.The constants are independent of n, u, and u0.Source: §2.7 · p. 10

3. Geometric Aspects of Domain Decomposition.

This section develops geometric and topological groundwork, defining subdomains and manifold-specific notions and proving facts used later. It also uses boundary-tracking sets to establish termination and membership in I_N, and states boundary properties of I_1 and I_2.

3.1. Interior and Boundary.

  • Manifold interior and boundary are intrinsic, unlike point-set interior and boundary, which depend on the ambient space; the paper uses the manifold meanings by default.For a manifold, the boundary consists of points locally modeled on a closed half-space, while its complement is the interior.Source: §3.1 · p. 11

3.2. Subdomains.

  • A subdomain is a connected, compact, same-dimensional embedded submanifold with boundary; the definition uses an extension of M across its boundary to handle boundary points.The extension is formed by gluing ∂M × [0, 1) to M, and embeddedness imposes a local half-space condition at boundary points.Source: §3.2 · pp. 11–12
  • The topological lemmas identify the point-set interior of Mi as Mi \ γi, give γi \ closure(γi) = γi ∩ ∂M, and show γi ∩ ∂M has no relative interior in ∂M.Source: §3.2 · pp. 13–14

3.3. Lipschitz Boundary.

  • A Lipschitz boundary is locally the graph of a Lipschitz function in a chart, and this property is independent of charts compatible through C1-diffeomorphisms.This regularity admits domains such as polyhedra with corners and is important for analysis, although the earlier topological lemmas do not require it.Source: §3.3 · pp. 14–16

3.4. Winding Number.

  • Lemma 3.6 gives inductive criteria for winding-number membership in I_n and for identifying new members of I_n \ I_n-1 using subdomain boundaries and intersections with earlier U_j.The winding-number results are used in the proof of Proposition 2.8.Source: §3.4 · p. 17
  • The sets J_n track U_j containing boundary points of D_{n−1}; their disjointness forces later J_n to be empty and yields D_n0=M.Source: §3.4 · p. 17
  • The argument descends through indices n1>n2>⋯>ns=1 to show i0∈I_s⊆I_N, even when s>N.Source: §3.4 · p. 17
  • The union of U_i for i∈I_1 is an open neighborhood of ∂M, and any M_i meeting ∂M has i∈I_2.Source: §3.4 · p. 17

4. Basic Theory of Elliptic Equations.

This section reviews elliptic theory on manifolds and develops the maximum-principle and Perron-solution results used in the convergence analysis.

4.1. Maximum Principle.

  • The strong maximum principle rules out a nonnegative interior maximum for Lv ≤ 0 unless v is constant, while comparison follows from operator and boundary inequalities.If boundary values differ, the comparison is strict in the interior.Source: §4.1 · pp. 18–19

4.2. Perron Solution.

  • The Perron solution is the supremum of subfunctions with prescribed boundary bounds and is bounded, smooth, and solves Lu = 0 in the interior.It respects ordered boundary data and is strictly ordered in the interior when the resulting solutions differ.Source: §4.2 · p. 19
  • For Lipschitz boundary and continuous boundary data, the Perron solution is continuous up to the boundary, attains the data, and is the unique continuous classical solution.At any boundary point where the data are continuous, the Perron solution is continuous there and matches the data.Source: §4.2 · p. 19

5. Convergence Rate.

This section characterizes the factors that bound the convergence rate and establishes how they respond to coefficient and decomposition changes.

Convergence-rate factors

  • 0 ≤ θ_i ≤ 1, and θ_i is positive in the interior; it is strictly below 1 under the stated boundary or coefficient conditions.In particular, c > 0 somewhere in M_i implies θ_i < 1 in M_i \ ∂M_i.Source: §5 · p. 19
  • 0 < Θ ≤ 1, and Θ < 1 if c > 0 somewhere in every M_i with ∂M_i = γ_i.Θ is formed from the component factors and their bounds on the supports of the partition functions.Source: §5 · p. 20

Coefficient comparison

  • Increasing c does not increase Θ, and as c tends to +∞, Θ tends to 0 uniformly on the supports of the partition functions.The uniform convergence follows from decreasing convergence of θ_i and compactness of supp ρ_i.Source: §5 · p. 21

Decomposition comparison

  • For the compared decompositions, Θ( eD) < Θ(D) when Θ(D) < 1.The strict inequality is obtained by comparing the corresponding θ_i functions.Source: §5 · pp. 21–22
  • A refinement makes Θ nondecreasing, while a strict refinement makes Θ strictly larger.The strict comparison is established for a point in the support of a partition function under the strict-refinement condition.Source: §5 · p. 22

6. C0-Convergence.

The section represents the iteration error by powers of a linear operator and proves geometric convergence with boundary, while also identifying conditions under which errors can retain their norm for several steps.

6.1. First Step.

  • For any initial guess, the error remains in C0_0(M) and evolves by u_n−u=T(u_{n−1}−u)=T^n(u_0−u).The operator T is linear and continuous, with ||T||≤Θ.Source: §6.1 · p. 23

6.2. With Boundary.

  • The strict bound T^nτ<1 for n≥N−1 is proved inductively, advancing from subdomains adjacent to the boundary toward the interior.The induction uses the strong maximum principle; Proposition 6.7 takes N to be the winding number.Source: §6.2 · pp. 24–25
  • With boundary, the iterates satisfy ||T^n||≤CL^n for some L in (0,1), giving a geometric convergence bound.The proof obtains strict contraction after a number of steps determined by the winding number N.Source: §6.2 · pp. 24–25
  • In the stated cases, some errors retain norm 1 through n≤N−1 or n≤N−2, so their norms need not decrease in the first steps.The later bound is n≤N−1 under the additional condition described in the proof.Source: §6.2 · pp. 25–26

7. High Regularity.

The section reduces manifold norm estimates to local coordinate estimates and applies elliptic regularity to control iteration errors. The resulting Ck,α estimate has a constant independent of the iteration index and the solutions.

7.1. Function Spaces.

  • On a compact manifold, a finite coordinate cover and subordinate partition of unity reduce Ck estimates to local Euclidean estimates.The partition-based norm is equivalent to the manifold norm.Source: §7.1 · pp. 26–27

7.2. Elliptic Regularity.

  • Elliptic regularity yields the local derivative estimates by induction, including at boundary portions where the local error vanishes.The proof handles interior and boundary portions and establishes the local estimate for general k.Source: §7.2 · pp. 28–29
  • The resulting Ck,α estimate has a constant independent of n, u, and u0.The constant depends on k, α, the supports suppρi, and the Ck,α norms of the partition functions.Source: §7.2 · p. 29
  • For Theorem 2.24, the local iteration error is identified with a continuous H1 solution of the auxiliary problem by uniqueness.The proof first obtains an H1 solution and uses boundary regularity to establish continuity.Source: §7.2 · p. 29

8. Generalization.

The generalization establishes C0 geometric convergence under Assumptions 2.1 and 2.3 alone, extends the operator class to all second-order linear elliptic equations on compact manifolds, and discusses lower-regularity coefficients. It also identifies how the convergence bounds and derivative convergence depend on the problem’s geometric and regularity conditions.

8.1. Convergence in General.

  • The generalized winding number eN(c, D, ρ) also applies when ∂M is empty and depends on c; when ∂M ≠ ∅ and c = 0, it equals N(D, ρ).If c is positive somewhere in every subdomain with ∂Mi = γi, then eN(c, D, ρ) = 1.Source: §8.1 · p. 31
  • Theorem 8.3 establishes full-general C0 geometric convergence under Assumptions 2.1 and 2.3, with its geometric bound independent of the iteration, initial guess, and solution.The error is also nonincreasing between iterations.Source: §8.1 · pp. 31–32
  • When eN > 2, a particular non-solution initial guess can retain its initial C0 error for n < eN − 2.The theorem also gives a boundary-condition-dependent extension of this interval.Source: §8.1 · p. 31

8.2. Equations of General Type.

  • On compact manifolds, the operator form used in the paper covers all second-order linear elliptic equations, and the theory carries over to that full class.The reduction uses a potentially different Riemannian metric while preserving the relevant uniform function-space structures and the condition on c.Source: §8.2 · pp. 32–33

8.3. Equations with Low Regularity.

  • The principal results remain valid for coefficients below C∞ regularity, provided they are not excessively singular, but the weak formulation requires sufficient regularity of the functions involved.For example, with g in C1 and b and c in C0, the text requires u in H1(M) and ρi in C0,1(M).Source: §8.3 · p. 33
  • With low-regularity coefficients, smooth convergence of un − u is not guaranteed, and derivative convergence depends on coefficient, data, and boundary regularity.Convergence may still hold in weak-to-strong senses up to a regularity-dependent order.Source: §8.3 · p. 34

9. Conclusion.

The conclusion presents a continuous domain decomposition method for elliptic boundary value problems on compact Riemannian manifolds and states that its operator class covers all second-order linear elliptic equations there. It highlights general C0 geometric convergence, rate estimates, and derivative convergence under suitable regularity assumptions.

9. Conclusion.

  • On compact manifolds, the paper’s elliptic problem covers the full class of second-order linear elliptic equations.Source: §9 · p. 34
  • The general Theorem 8.3 characterizes C0 convergence under Assumptions 2.1 and 2.3 and establishes geometric convergence for Algorithm 2.1.The conclusion also describes detailed proofs of three special cases.Source: §9 · p. 34
  • Theorems 2.6, 2.17, and 2.20 estimate C0 convergence rates and show how domain-decomposition geometry and equation coefficients affect convergence.Source: §9 · p. 34
  • Under suitable regularity assumptions, C0 convergence implies convergence of high-order derivatives.Source: §9 · p. 34
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