Diffraction of waves by finite thin coatings: asymptotic expansion and effective models

• The document is a doctoral thesis prepared at École nationale supérieure de techniques avancées, within Institut Polytechnique de Paris.Source: NNT : 2026IPPAE022 · p. 1
• Its doctoral specialty is applied mathematics, and it belongs to the Hadamard Mathematics Doctoral School (EDMH).Source: NNT : 2026IPPAE022 · p. 1
• The thesis was presented and defended at Palaiseau on September 29, 2026.Source: NNT : 2026IPPAE022 · p. 1

C´EDRIC BAUDET

• The jury includes a president, rapporteur, examiners, thesis supervisors, and an invited member.The listed roles include president, rapporteur, examiner, thesis director and co-director, and guest.Source: C´EDRIC BAUDET · p. 1

Remerciements

• The author thanks supervisors Sonia and Patrick for patient guidance, availability, autonomy, and support in developing and presenting the work.Source: Remerciements · p. 2
• The acknowledgments also thank jury members, laboratory colleagues, fellow doctoral students, friends, and family for scientific discussions, practical help, and personal support.Source: Remerciements · pp. 2–3
• The listed Chapter 9 topics include effective-model well-posedness, convergence and error estimates, and numerical results.Source: Remerciements · p. 6

1.1 Motivations and setting of the problem

• The study treats a two-dimensional semi-infinite layer ending at a corner of arbitrary angle, possibly with a local geometric perturbation at the layer-thickness scale.Fine meshes make thin-layer problems costly when coating thickness is small relative to the wavelength and object size.Source: §1.1 · p. 7

1.2 State of the art

• For finite-layer Helmholtz problems, prior expansions were limited to order 2, while high-order effective models remained open and corner singularities complicate boundary conditions.The work addresses the expansion challenge with algebraic calculus tools that make higher-order matching conditions more explicit.Source: §1.2 · p. 11

1.3 Summary of the thesis

• The thesis develops an all-orders expansion, examines its properties and numerical validity, and derives effective models of orders 3 and 5.The expansion and numerical validation are treated in Chapters 3–8; Chapter 9 presents the effective models.Source: §1.3 · p. 12

1.3.1 Presentation of the matched asymptotic expansion

• The expansion uses matched asymptotics, dividing the domain into far, layer, and corner zones with separate fields linked by matching conditions.The fields are constructed inductively in powers of ε and ln ε, building the layer and far fields alongside the corner fields.Source: §1.3.1 · pp. 12–13
• Matching links the corner behavior of the far and layer fields to the corner fields’ behavior at infinity, requiring singular parts beyond natural variational spaces.These singular parts are chosen to satisfy matching conditions and depend on preceding fields’ asymptotic behavior.Source: §1.3.1 · pp. 13–14

1.3.2 Chapters 3 and 4: the matching conditions

• Explicit singularity bases, operators on singularities, and formal series yield matching conditions at all orders at once.The formal series permit non-integer powers, while the matching coefficients depend on the angle and material parameters, not the source or local corner geometry.Source: §1.3.2 · pp. 15–17

1.3.3 Chapter 5: construction of the expansion

• Chapter 5 establishes existence and uniqueness of the expansion fields and rigorously justifies the asymptotic expansion with a sharp error estimate.The chapter also gives a way to construct the far fields directly by eliminating the layer and corner fields.Source: §1.3.3 · pp. 17–18

1.3.4 Chapters 6 and 7: additional properties of the expansion

• For irrational Θ/π, or rational Θ/π when p is below its denominator, the expansion through order p has no ln ε and the matching conditions simplify to one-to-one relations.At order 5, the stated exceptional angles are 2π/3, π, 4π/3, and 5π/3; the method itself applies at any angle and order.Source: §1.3.4 · p. 19
• Symbolic examples show angle-dependent simplifications and complexity, while genericity characterizes fields or coefficients that vanish only at exceptional parameter values.The chapter also establishes local convergence of the modal series and uniform convergence on smaller balls.Source: §1.3.4 · p. 20
• The far-field modal decomposition converges classically, while weighted norms on formal series of corner singularities establish convergence and bounds for the expansion profiles.The norms make a series finite precisely when it converges, with weights indicating convergence speed; continuity of the singularity operators supports the proof.Source: §1.3.4 · p. 20

1.3.5 Chapter 8: numerical validation

• Relative errors in computed corner-profile coefficients range from roughly 10^-5 to 10^-10 for a domain with L = 10, and numerical results validate the expansion through order 5.The enriched Galerkin method uses complements at infinity; the authors describe the order-5 validation as the first numerical computation beyond order 2 for a corner and thin-layer problem.Source: §1.3.5 · pp. 21–22

1.3.6 Chapter 9: effective models

• Order-3 and order-5 effective models avoid fine meshing of the layer and corner vicinity, are well-posed and uniformly stable for sufficiently small ε, and have proved error estimates.The models use effective boundary conditions on Γ◦ and a corner DtN condition on an artificial boundary; the order-5 model also imposes a non-classical DtN condition at the tip. Numerical results show their efficiency.Source: §1.3.6 · pp. 22–23

2.1 Motivations et description du problème

• L’étude porte sur un problème bidimensionnel de revêtement partiel, modélisé près de son bord par une couche rectiligne semi-infinie et un coin d’angle arbitraire, éventuellement perturbé localement à l’échelle de l’épaisseur.Le maillage fin de la couche est coûteux lorsque son épaisseur est très faible ; l’analyse suppose aussi Im(ω) ≠ 0 dans ce cadre académique semi-infini.Source: §2.1 · pp. 25, 27

2.2 État de l’art

• Pour Helmholtz avec couche finie, les développements antérieurs étaient limités à l’ordre 2 ; cette thèse vise tout ordre et tout angle grâce à des outils algébriques.Les singularités de coin s’organisent en arbre de dépendances, ce qui complique la construction à tout ordre et la justification des erreurs optimales.Source: §2.2 · p. 29

2.3 Résumé de la thèse

• La thèse construit un développement asymptotique à tout ordre aux chapitres 3 à 5, en étudie les propriétés et la validité numérique aux chapitres 6 à 8, puis en déduit des modèles effectifs d’ordres 3 et 5 au chapitre 9.Les chapitres 4 et 5 construisent récursivement les champs lointains, de couche et de coin.Source: §2.3 · p. 30

2.3.1 Présentation du développement asymptotique raccordé

• Le développement raccordé distingue trois zones — couche, voisinage du coin et reste du domaine — avec des champs lointains, de couche et de coin développés en puissances de ε et de ln ε.La construction récursive par ordre p ne laisse qu’un nombre fini de termes non nuls ; les champs sont liés par des conditions de raccord.Source: §2.3.1 · pp. 30–31
• Les comportements des champs lointains et de couche au coin doivent se raccorder à ceux des champs de coin à l’infini, malgré leurs singularités.Ces singularités imposent de spécifier les parties non variationnelles des champs plutôt que de les chercher dans leurs espaces variationnels naturels.Source: §2.3.1 · p. 32

2.3.2 Chapitres 3 et 4 : les conditions de raccord

• Une base explicite des singularités, des opérateurs qui les relient et des séries formelles permettent de traiter la complexité du raccordement.Les séries formelles utilisées acceptent des indices réels, afin de représenter les puissances non entières de ε et de r.Source: §2.3.2 · pp. 34–36
• Les conditions de raccordement caractérisent l’hypothèse de raccord à tous les ordres et constituent le résultat principal du chapitre 4.Les coefficients de raccord ne dépendent que de l’angle, des paramètres du milieu et de la fréquence.Source: §2.3.2 · p. 35

2.3.3 Chapitre 5 : construction du développement

• Le chapitre 5 établit l’existence et l’unicité des champs du développement, propose une construction directe des champs lointains et justifie l’expansion par des estimations d’erreur.Les champs de couche sont exprimés en fonction des champs lointains, et des problèmes modèles construisent les champs lointains et de coin.Source: §2.3.3 · pp. 36–37
• Le chapitre 5 conclut la construction du développement présentée dans les chapitres 3 à 5.L’extrait signale le théorème 5.4.5, mais ne fournit pas son énoncé complet.Source: §2.3.3 · p. 37

2.3.4 Chapitres 6 et 7 : propriétés additionnelles du développement

• Dans la plupart des cas, si Θ/π est irrationnel, ou rationnel avec p inférieur au dénominateur de Θ/π sous forme irréductible, le développement de uε jusqu’à l’ordre p ne contient pas de ln ε et la condition de coin se simplifie substantiellement.Source: §2.3.4 · pp. 37–38
• Les séries formelles des champs lointains convergent, et leur décomposition modale est aussi valable au sens classique, ce qui justifie leur calcul numérique et soutient les modèles effectifs.Dans la plupart des cas, une représentation par fonctions de Bessel explique l’absence de termes en ln r et les simplifications associées; les normes pondérées servent à établir la convergence des séries.Source: §2.3.4 · pp. 38–39

2.3.5 Chapitre 8 : validation numérique

• Pour les correcteurs de coin, la méthode de Galerkine enrichie de compléments à l’infini est d’ordre élevé : plus N est grand, meilleur est le taux de convergence; l’erreur relative sur les coefficients va de 10−5 à 10−10, même pour un domaine ΩL de taille raisonnable (L = 10).Source: §2.3.5 · p. 40
• Le développement asymptotique est validé numériquement jusqu’à l’ordre 5, pour la première fois au-delà de l’ordre 2 pour un problème avec angle et couche mince.Le calcul des champs lointains traite leurs singularités en introduisant une frontière artificielle autour du coin et une condition de Dirichlet-to-Neumann non homogène.Source: §2.3.5 · pp. 40–41

2.3.6 Chapitre 9 : modèles effectifs

• Les modèles effectifs d’ordres 3 et 5 évitent de mailler finement la couche et le voisinage du coin, en associant des conditions effectives sur Γ◦ à une condition DtN sur Σ◦.Le modèle d’ordre 5 impose aussi une condition DtN non classique à l’extrémité P◦; ses opérateurs DtN reposent sur une décomposition modale tronquée intégrant les singularités et correcteurs de coin.Source: §2.3.6 · pp. 41–42
• Les modèles effectifs sont bien posés et uniformément stables pour ε suffisamment petit, et une estimation d’erreur est établie, avec un exposant ℓ_m dépendant de Θ et nul pour la plupart des angles.Les passages fournis ne reproduisent pas la formule complète de stabilité ni celle de l’estimation d’erreur.Source: §2.3.6 · p. 42
• The convergence order can be improved if ∂nu^(m)_ε is sufficiently regular, although elliptic regularity arguments are complicated by the corner at P◦.Source: §2.3.6 · p. 42
• For Θ = π and ε = 0.1, figures 2.8–2.10 show the exact solution and effective-model errors; numerical results are also provided for other relevant angles.The text identifies Θ = π as particularly relevant because its asymptotic developments are among the most complex, and says the numerical results show the models’ efficacy.Source: §2.3.6 · p. 43, Figure 2.8 · p. 43, Figure 2.9 · p. 43

3.1 Preliminary: an introduction to the usual algebraic formal series

• Formal series extend convergent power series to arbitrary coefficient sequences that can be manipulated algebraically even when they do not define functions.The section introduces this framework to prepare for formal series of corner singularities used to derive matching conditions.Source: §3.1 · p. 44
• For smooth functions, formal series encode asymptotic behavior without requiring convergence; corner far-field expansions are built formally, with convergence proved later in Chapter 7.This avoids the need to handle truncation and small-o terms while computing asymptotic behavior.Source: §3.1 · p. 45, §3.1.1 · p. 46

3.1.1 Definition of the formal series set C[[X]]

• C[[X]] is defined as C^N, with scalar multiplication, addition, and Cauchy-product multiplication that align with limit functions and Taylor or asymptotic expansions when defined.The coefficients of a smooth function’s formal series represent its asymptotic behavior at zero.Source: §3.1.1 · pp. 45–46
• A family of formal series has finite support on each coordinate when, for every n, only finitely many coefficients a_i,n are nonzero, making their coefficient sum a classical finite sum.Source: §3.1.1 · p. 47
• Formal series encode coefficient sequences, may converge or diverge, represent smooth-function behavior at zero, and permit sum, product, and composition when each coefficient uses finitely many nonzero terms.These operations make them manipulable like power series without requiring convergence.Source: §3.1.1 · p. 50

3.1.2 Examples of problems solved in C[[X]]

• Formal series simplify asymptotic calculations by enabling algebraic change-of-variable operations and operator inversion for differential equations.In the example, id − A is inverted by the series sum of A^n; the resulting series may diverge, as for g∞ = X.Source: §3.1.2 · pp. 51, 54

3.2 A quick review of existing generalized algebraic formal series

• The proposed generalized series use vector-space terms without multiplication, allow index sets not bounded below or even ordered, and support a new summation operation for infinite series.The authors also introduce degree-based operators and calculational tools for summing and inverting operator series.Source: §3.2 · p. 55
• Formal series in T represent powers of ln ϵ as polynomials with coefficients in a vector space E.The construction uses finite-support sequences, with T acting by shifting their indices.Source: §3.2 · pp. 55–56

3.3 General definitions

• Formal series over indexed vector spaces use locally finite support, while sums of series are defined when their family has collectively locally finite support.These series have coefficientwise vector-space operations, but no multiplication is defined because it is not needed here.Source: §3.3 · p. 56, §3.4 · p. 59

3.4 Formal series with index in R

• Real-indexed formal series represent corner singularities that are homogeneous up to logarithms, with locally finite support ensuring their components can be summed as formal series.The component spaces are assumed to form a direct sum, giving each element a unique component family.Source: §3.4 · pp. 59–60
• Section 3.4 characterizes formal series indexed by R through families with locally finite support and describes inclusion relationships among these spaces.The text notes that these inclusions are generally strict.Source: §3.4 · p. 61

3.4.1 Operators with a degree

• Degree-tracking operators extend to formal series, with degrees adding under composition and operators commuting with locally finite formal sums.The proposition applies when the family of formal series has collectively locally finite support.Source: §3.4.1 · pp. 62–63

3.4.2 Geometric series of operators

• For operators built from finitely many positive- or negative-degree maps, id − f is invertible on E+ or E− via a Neumann series, enabling formal-series equations for corner-field behavior.The resulting operator expressions can be complex, but the calculus avoids needing to specify their individual terms.Source: §3.4.2 · p. 66

3.5 Formal series with index in R2 (involving ϵ)

• Section 3.5 defines formal series indexed by R2 and involving ϵ to represent the behavior of asymptotic sums.Source: §3.5 · p. 67
• R2-indexed formal series encode the joint asymptotic behavior in thickness and radius, with thickness powers that need not be integers.The framework focuses on powers of ϵ; powers of ln ϵ are handled by the earlier definition.Source: §3.5 · pp. 67–68

3.5.1 Multiplication by powers of ϵ

• Multiplication by ϵ^p shifts the thickness-power index by p and defines a group action of the additive real numbers.The shift preserves locally finite support, allowing formal series to be written in powers of ϵ.Source: §3.5.1 · p. 69

3.5.2 Operators with a degree

• Operators with a degree extend to the R2-indexed series, with the extended degree recording both the original degree and the ϵ-power shift.The degree is defined as the pair (deg f, p).Source: §3.5.2 · p. 71

3.5.3 Geometric series of operators

• When each operator in a finite family has positive degree along a direction v, the geometric series converges on Ev and gives the inverse of id − f there.The inverse is the Neumann series ∑ from n=0 to ∞ of f^n, under the stated positive-direction degree condition.Source: §3.5.3 · p. 72

Derivation of the matching conditions

• The matching assumption is equivalent to conditions linking corner-field coefficients on basis singularities, derived using explicit singularities, operators, and formal series.The chapter gives a necessary and sufficient condition for the matching assumption, assuming the fields exist and satisfy the other stated conditions.Source: Derivation of the matching conditions · p. 75

4.1.1 Definition of the spaces A

• The spaces A contain quasi-homogeneous corner singularities and have explicit bases; their definitions incorporate the Dirichlet boundary condition.Their elements behave like r^d ln^j r, and the spaces are built from explicit functions.Source: Derivation of the matching conditions · p. 75, §4.1.1 · p. 76

4.1.2 Solving the Poisson and Helmholtz equations in the spaces A

• The A framework supports canonical Poisson- and Helmholtz-like problems in the layer domain and enables an exact, fast, memory-thrifty solver.The chapter uses these tools to express field behavior around the corner and compute matching coefficients.Source: Derivation of the matching conditions · p. 75, §4.1.2 · pp. 79–80
• In A(Ω), Poisson problems with interior data and harmonic problems with Dirichlet data on Γ have unique solutions; their solution maps have degrees 2 and 0, respectively.The Dirichlet solution map is linear.Source: §4.1.2 · p. 81
• In A⊥(Π), three Poisson-type problems have unique solutions: with interior data, layer data, or prescribed normal derivative on Γ; the last solution map has degree 0.The normal-derivative problem’s solution map is linear.Source: §4.1.2 · p. 82
• The spaces A(Π) and A(Ω) are stable under the stated boundary normal-derivative and Laplacian operators, which map with degrees −1 and −2, respectively.The Laplacian maps A(Ω) and A(Π) to A(Ω).Source: §4.1.2 · p. 83

4.2 Formal series of corner singularities

• Formal-series spaces built from corner singularities are defined for the far, layer, and corner fields, with well-posed Helmholtz- and Poisson-like problems in those spaces.These tools are applied to derive matching conditions.Source: §4.2 · p. 84

4.2.1 Asymptotic behaviors w.r.t. r of solutions of model problems

• A+(D) and A−(D) represent behaviors involving increasing and decreasing powers of r, respectively, but their formal series may diverge and need not be functions.The increasing-power series describe behavior as r tends to zero; the decreasing-power series describe behavior as r tends to infinity.Source: §4.2.1 · p. 84
• Helmholtz and Poisson-like problems have unique solutions in A+(Ω), A+(Π), and A−(Π), and formal operator inversion is a key benefit of the algebraic approach.Truncations of the formal series give asymptotic expansions of the associated fields at a given order.Source: §4.2.1 · pp. 85, 87
• Matching conditions link the corner behavior of the far-and-layer fields to the behavior at infinity of corner fields, with some singularity coefficients chosen to enforce that match.The formal families are assumed to exist here; coefficients associated with non-variational modes can be fixed freely, and Section 4.3 determines how to choose them for matching.Source: §4.2.1 · p. 88

4.2.2 Formal series of corner singularities involving ϵ

• The framework rigorously defines formal corner-singularity series in ϵ and ln ϵ, extending singularity coefficient extractors and degree-based operators to those series.Two model problems in the formal-series spaces correspond to the layer-thickness-one problem and the rescaled exact solution, respectively.Source: §4.2.2 · pp. 88, 89, 90, 92

4.2.3 Scaling by ϵ of formal series of corner singularites

• The scaling operator Hϵ rigorously formalizes scaling despite ϵ being an indeterminate, is invertible, and maps A+ϵ series to A−ϵ series.Its definition expands logarithms such as ln(ϵx) into ln ϵ + ln x.Source: §4.2.3 · pp. 93–94
• Formal scaling H_ε is invertible and reproduces expected scaling properties, but generally does not commute with operators depending on the linear forms σ_d.Powers of ln ε are needed in the ansatz because scaling corner terms involving ln r naturally produces them.Source: §4.2.3 · pp. 94, 98

4.3 Equivalence between the matching assumption and the matching conditions

• The matching assumption is equivalent to scalar equations on the coefficients σ_d(u₀_p,ℓ), which are used to construct the fields by induction.These equations are called the matching conditions.Source: §4.3 · p. 99
• The matching ansatz restricts the components of u_p,ℓ and S_p,ℓ to specified degree sets and fixes summation indices, though it can be avoided with formal series of unknown locally finite support.The required a priori information is the form of the corner singularities, including logarithms of r and ε.Source: §4.3 · p. 99
• Theorem 4.3.3 factors corner behavior into a geometric series of operators encoding singularities and coefficients σ_d that must be fixed for the fields to match.This separates the singularity expressions from the matching task; the operators R_Δ, R_∂²Y, and R_N provide a compact description of the corner singularities.Source: §4.3 · p. 100
• Under Ansatze 4.2.7 and 4.3.2, the matching assumption is equivalent to the matching conditions, which express it through scalar relations on field coefficients.The derivation first shows the projected relations are necessary and sufficient, then re-expresses them as coefficient relations.Source: §4.3 · pp. 101–103
• The matching conditions have a convolutive structure in p and ℓ, and the field construction uses the subset that determines singular coefficients inductively.These are the relations for σ_d(u_0^{p,ℓ}) when d < 0 and σ_d(S^{p,ℓ}) when d > 0, corresponding to corner and infinity singularities, respectively.Source: §4.3 · pp. 102, 104

4.4 Summary of the method

• The derivation method builds a canonical problem, quasi-homogeneous basis functions and a singularity space, then projects field matching onto basis coefficients to obtain scalar relations.The method retains the relations useful for constructing the fields.Source: §4.4 · p. 105
• Problem-specific details are concentrated in the first three derivation steps, while the later matching reasoning is less dependent on the physical setting and reduces case-specific calculations.The approach may need adaptation when a common canonical problem for near and far fields cannot be derived, as in some high-contrast settings.Source: §4.4 · pp. 105–106
• For an infinite periodic layer, matching uses the vertical second-derivative operator; the preceding far-field trace determines near-field behavior at infinity, while the next trace supplies a Dirichlet condition for the far field.The example is presented as an illustration of the method’s generality and comparison with prior work.Source: §4.4 · pp. 106–107

4.5 Numerical computation of the fields behaviors and the matching coefficients

• A purpose-built symbolic module in C++ represents the mathematical spaces and operators, while Ratio and DValue types preserve exact rational operations when possible and avoid angle-classification errors from floating-point roundoff.DValue falls back to floating-point evaluation when exact operations are not possible.Source: §4.5 · p. 108
• Finite-order computations require only truncated formal series: each field and matching coefficient depends on a finite truncation, with operator-degree rules determining how far to truncate.The implementation also uses long-double complex arithmetic to improve accuracy while keeping computations fast and memory-thrifty.Source: §4.5 · p. 109
• Matching-coefficient computation uses layered classes built from simpler layers to manipulate typically tens or hundreds of numbers.Figure 4.6 schematically illustrates the classes; the bottom layer consists of integers and floating-point numbers.Source: §4.5 · pp. 112–113, Figure 4.6 · p. 113
• These computations are faster and more memory-thrifty than computing singularities by solving ODEs with respect to θ, for example with finite elements.Source: §4.5 · p. 112
• The computations are exact up to round-off errors, whereas finite elements require a fine mesh to achieve high precision.Source: §4.5 · p. 112
• The refinements to the method’s algorithms and optimal truncation orders are not critical to its efficiency.Source: §4.5 · p. 112

5.1 Existence and uniqueness for model problems

• To accommodate singularities at the corner and infinity, the construction augments natural variational spaces with truncated algebraic singularity spaces and establishes well-posedness for model problems in the resulting spaces.The truncation is near zero for far fields and near infinity for corner fields.Source: §5.1 · p. 115

5.1.1 Well-posedness of far fields-like problems

• The far-fields-like problem has a unique solution in H1(Ω) + χ0A(Ω) for the stated forcing, boundary data, and finitely supported prescribed singularity coefficients.Coefficients σd(u) with d < 0 can be prescribed because they describe non-variational corner behavior; coefficients with d > 0 belong to the variational part.Source: §5.1.1 · pp. 116–117

5.1.2 Well-posedness of corner fields-like problems

• The corner-fields-like problem has a unique solution in V + χ∞A(Π) for the stated weighted source, admissible profile, and finitely supported prescribed singularity coefficients.The profile may be discontinuous across Γ.Source: §5.1.2 · p. 119

5.2.1 Statement of the results

• Solutions of the model problems admit asymptotic expansions in A±(Π), describing far- and layer-field behavior as r → 0 and corner-field behavior as r → ∞.These expansions are used to inductively build up,ℓ, Up,ℓ, and Sp,ℓ and describe their asymptotic behavior.Source: §5.2.1 · p. 120
• The expansion framework uses differentiable small-o remainders so asymptotic behavior remains valid under the derivatives needed in the inductive field equations.The fields depend on preceding fields and their derivatives.Source: §5.2.1 · p. 120
• Asymptotic behaviors are unique, and differential operators of degree d map an established behavior to the corresponding behavior shifted by d.The operator result states that δ(f) has the behavior obtained by applying δ to the asymptotic profile; uniqueness is asserted for both r → 0 and r → ∞.Source: §5.2.1 · p. 121
• The far-and-layer problem admits an expansion at the corner, with its local field decomposed into a corner profile and a regular part under the stated data assumptions.The forcing vanishes near the corner, while the layer and interface data have prescribed asymptotic behaviors; the solution has an asymptotic expansion to every order as r → 0.Source: §5.2.1 · pp. 121–122
• The corner field and its source terms admit all-order asymptotic expansions as r → ∞ in the outer domain and layer.The theorem states expansions for the restrictions of F to the outer domain and layer, as well as for S.Source: §5.2.1 · p. 122

5.2.2 Proof of Theorem 5.2.5: asymptotic behavior for corner fields-like problems

• The corner proof uses Kondrat’ev theory and a Laplace transform in logarithmic coordinates to extend the transformed field meromorphically, with simple poles identifying expansion terms.The inverse operator has simple poles at π/Θ Z*, and the residue theorem extracts terms; iterating this procedure yields an expansion at any order.Source: §5.2.2 · pp. 122, 124, 125
• For every d, the corner field S has the asymptotic form S = T⩾d(S∞) + o∂(r^d) as r → ∞.The proof inductively determines coefficients of the asymptotic terms and controls the remainder.Source: §5.2.2 · pp. 129–130

5.2.3 Proof of Theorem 5.2.4: asymptotic behavior for far-and-layer fields-like problems

• For far-and-layer fields, the proof treats the asymptotic expansion as r → 0 using a cutoff near the origin and a Laplace transform extended from a left half-plane.Propositions 5.2.10 and 5.2.12 provide the ingredients used to prove Theorem 5.2.4.Source: §5.2.3 · pp. 130, 133

5.3.1 Inductive construction of the fields

• The inductive construction summarizes how to build the fields u_p,ℓ, U_p,ℓ, and S_p,ℓ, using cutoffs localized near the origin and infinity.It recalls the coefficient definitions and spaces used in that construction.Source: §5.3.1 · p. 133
• The inductively constructed fields are well-defined and satisfy the required inductive relations, asymptotic behaviors, and regularity on Γ.The regularity property on Γ is needed for the subsequent error-estimate proof.Source: §5.3.1 · p. 134
• For each p, the fields u_p,ℓ, U_p,ℓ, and S_p,ℓ vanish for all sufficiently large ℓ, with the threshold depending only on Θ and p.The proposition states that the threshold does not depend on the material, frequency, domain, or source terms.Source: §5.3.1 · p. 135
• The construction satisfies Ansatz 4.3.2 and the matching assumption of Definition 4.3.1.The proof verifies the matching conditions using the inductive definitions and Lemma 5.3.3.Source: §5.3.1 · p. 136

5.3.2 Practical way to build the far fields

• Far fields can be built directly without computing layer or corner fields, replacing the layer with boundary conditions and using corner conditions that depend on corner profiles.The layer boundary conditions are identical to those for an infinite layer.Source: §5.3.2 · pp. 136, 138
• The far-field construction admits a small improvement useful for effective models, but no analogous improvement is available for corner profiles.For angles satisfying the example’s condition, the first nonzero far fields are u0,0 and u1,0.Source: §5.3.2 · p. 139

5.4 Error estimates

• The approximate global field is assembled from far-and-layer fields with a cutoff, yielding a piecewise-defined approximation on the perturbed domain.The construction is stated for any sufficiently small ε and truncation order P.Source: §5.4 · p. 140

5.4.1 Estimation of the matching error

• Choosing η = √ε makes the matching zone shrink toward the far-field scale while expanding toward infinity on the corner-field scale.A matching-error lemma then bounds the discrepancy in that zone, although its bound is not legible in the supplied passage.Source: §5.4.1 · p. 140

5.4.2 Global and far fields error estimates

• The global H1 error is improved to o(ε^(P/2)), but achieving an ε^P far-field error requires an expansion through order 2P.The stated far-field error estimate applies when ε < δ^2; this condition reflects that the approximation consists only of far fields outside the matching zone.Source: §5.4.2 · pp. 143–144

6.1 Expression of far fields singularities using Bessel functions

• Far-field corner singularities are expressed using Bessel functions, aiding their numerical computation, clarifying the expansion’s structure, and supporting the later far-field modal decomposition.For negative integer orders, the usual first-kind Bessel-function representation may not have the required behavior near the corner; the appendix gives an extension using second-kind Bessel functions.Source: §6.1 · pp. 146, 148, 152
• When Θ/π is irrational, higher-order singularities have Bessel-function representations with coefficients supported on [-n,n], symmetric under index reversal and restricted to indices of parity n.For n > 0, the coefficient at index 0 vanishes.Source: §6.1 · p. 150
• For rational Θ/π = a/b, the same representation applies for n < b when d is not a negative integer; excluded cases can require second-kind Bessel functions and include logarithmic terms.The appendix notes that the bound n < b is not optimal in some special cases, including b ∈ {2, 4}.Source: §6.1 · pp. 150, 152

6.2 Simplifications depending on the angle

• For irrational Θ/π, no ln ε terms appear in the expansion; rational angles behave similarly at small orders.These angle-based simplifications are developed using the properties of the corner profiles Φ^0_{d,n} to study matching coefficients.Source: §6.2 · p. 152
• The equations for Φn have the same structure as those for un,0, and uniqueness identifies Φn with Φ0.The layer restriction Φn|Λ can be expressed using lower-order normal derivatives on Γ.Source: §6.2 · p. 152

6.2.1 The irrational case

• For irrational Θ, the expansion has no powers of ln ε, and the corner fields have no logarithmic terms in r or x.The authors attribute the Helmholtz result to cancellations tied to corner profiles expressed as linear combinations of Bessel functions of the first kind.Source: §6.2.1 · pp. 153–155

6.2.2 The rational case

• For rational Θ, the expansion of uε contains no ln ε before order pmin.The expansion behaves like the irrational case at the first orders.Source: §6.2.2 · pp. 155, 157
• Except for a few angles, the rational-angle expansion contains no ln ε at orders p < 5.The passage names Θ ∈ {π, 2π/3, 4π/3, 5π/3} as exceptions; it also notes that some pmin values are not optimal.Source: §6.2.2 · p. 158
• Proposition 6.2.6 partially describes the coefficients cᵤ←S; a stated case gives coefficient 1 when (d′, p) is (d, −d) or (d, d), with ℓ = 0.The passage presents these as alternative cases, respectively.Source: §6.2.2 · p. 158
• In the rational case, logarithmic corner terms do not match automatically with far-field terms, so specific matching relations are needed and can be difficult to compute because they depend strongly on Θ.The section also distinguishes matching-relation types for different cases.Source: §6.2.2 · p. 158

6.3.1 Genericity

• At fixed Θ, expansion structure is largely angle-driven, and generic nonzero far-field orders range from initially integer orders at small angles to many noninteger orders at larger angles.A quantity is generically nonzero when its zero set in the governing parameters has measure zero; the rectangular layer at Θ = π/2 is a special case without ln ε, although logarithms occur generically at that angle.Source: §6.3.1 · pp. 159–161, Figure 6.3 · p. 162

6.3.2 Detailed examples for a few angles

• The examples show that irrational-angle cases Θ = 2.5 and Θ = 4 have no ln ε and matching coefficients c_u←S and c_S←u equal 0 or 1, while Θ = 4 brings an earlier corner condition and a heavier expansion.The first corner condition appears at p ≈ 1.57 for Θ = 4 versus p ≈ 2.51 for Θ = 2.5.Source: §6.3.2 · p. 162
• For Θ = π, ln ε appears at p = 2 and coefficients are complex even at small orders, while the generically nonzero far-field orders in the example are rational, even integers.The examples enumerate far fields, corner profiles, matching coefficients, and corner conditions up to order p = 4 for three angles; their tables were generated by the symbolic module.Source: §6.3 · p. 159, §6.3.2 · pp. 162–163
• Detailed examples tabulate non-zero coefficients, corner-profile coefficients, and far fields with p ≤ 4 for Θ = 4 and Θ = π.The far-field lists are described as generic; the supplied passages do not provide their individual values.Source: Table 6.3 · p. 163, Table 6.4 · p. 164, Table 6.5 · p. 164

6.3.3 Synthetic representation of properties of the expansion for various orders and angles

• Computational-cost curves increase with order and generally lie higher for larger angles, with exceptions including crossings between Θ = 4 and Θ = 3π/2 and between Θ = 3 and Θ = π.At Θ = 4, more far fields are needed than at Θ = 3π/2 because distinct order-generating integer pairs can coincide for the rational angle; at Θ = π, logarithmic terms can require multiple far fields at an order.Source: §6.3.3 · p. 168
• For rational angles with denominator at most 6, the first logarithmic order is tabulated; the theorem’s bound appears optimal for odd denominators but not even ones.For larger denominators, the theorem excludes logarithms before order 7; the authors conjecture similar techniques could explain non-optimality for other even denominators.Source: §6.3.3 · p. 168
• The maximal logarithmic power starts at zero for every plotted angle, since the first expansion terms contain no ln ε.Figure 6.7 tracks this maximum against convergence order p from 0 to 10.Source: §6.3.3 · p. 168, Figure 6.7 · p. 170

• The appendix studies series Φ0_d,n that can be expressed using Bessel functions of the second kind.Source: 6.A Appendix: extension of the results… · p. 171
• It improves Theorem 6.2.5 for some angles at orders relevant to the effective models.Source: 6.A Appendix: extension of the results… · p. 171
• The results extend Sections 6.1 and 6.2.2 to particular cases, with proofs following reasoning similar to those sections.Source: 6.A Appendix: extension of the results… · p. 171

Extension of Section 6.1

• For negative integer indices, Bessel functions of the second kind fill the gap because Ya behaves like r^a in those cases.Source: Extension of Section 6.1 · p. 171
• The identities and differential-equation properties established for Ja are extended to Ya and the associated angular formal series.Source: Extension of Section 6.1 · p. 172
• Combining the second-kind Bessel results with earlier propositions gives explicit expressions for Φ0_d,n.Source: Extension of Section 6.1 · p. 173
• For rational angles Θ = aπ/b, the extension treats selected cases beyond the earlier n range, including b = 2 and b = 4.Source: Extension of Section 6.1 · pp. 173–174
• For b = 2 or 4, the construction of order-3 and order-5 effective models is as simple as for π/Θ irrational, unlike the more complicated cases b = 1 or 3.Source: Extension of Section 6.1 · p. 174
• The calculations cover b = 2 and 4 only through n ≤ 4, and the authors leave a possible broader pattern for even b uninvestigated.Source: Extension of Section 6.1 · p. 174

Extension of Section 6.2.2

• β is the largest index range for which Φ0_d,n remains a linear combination of first-kind Bessel profiles for the specified d values.Source: Extension of Section 6.2.2 · p. 175
• The matching coefficient identities hold under the stated bounds, including p + d′ < β and p + d < −2d′, or p − d < β and p + d′ < 0.Source: Extension of Section 6.2.2 · p. 175
• The generalized results relax the earlier assumptions d′ > −b and p < b.Source: Extension of Section 6.2.2 · p. 176
• For b not in {1, 3}, the improved range satisfies pmin ≥ 5, with the gain especially notable for b = 2 or 4.Source: Extension of Section 6.2.2 · p. 177

7.1 Statement of the results

• The decomposition advances effective models and justifies numerical computation of the far fields by induction.Summing decompositions of the first far fields is intended to approximate the exact solution near the layer using a Dirichlet-to-Neumann operator.Source: §7.1 · p. 178
• The far fields admit a modal decomposition with uniform convergence inside the source-free radius Rs; its coefficients are matching quantities σd(up,ℓ).The decomposition is valid on Ω∩B(0, Rs), and the coefficient bound depends on any chosen R in (0, Rs).Source: §7.1 · pp. 178–179
• The remainder estimate gives geometric convergence as the truncation order N increases, while Rs is the optimal radius of convergence.The estimate also contains a logarithmic factor relevant when π/Θ is rational; its power is generally not optimal.Source: §7.1 · p. 179

7.2.1 The rational case: estimates on individual terms

• For the rational case, coefficient-sum norms and restricted subspaces control exponents and logarithmic polynomial degrees, enabling bounds for the operators used to estimate individual terms.The restrictions support continuity estimates for the operators and prevent their constants from growing uncontrollably at high order.Source: §7.2.1 · pp. 181–182
• For s > 1 and d ≥ δ, the weighted norm satisfies ∥∂xφ∥γ ≤ Cδ,n,s s^d ∥φ∥γ for φ ∈ A^n_d(Γ).The constants are independent of d.Source: §7.2.1 · pp. 184–185

7.2.2 The rational case: estimates on formal series

• For s′ > s > 0, formal-series differentiation is bounded: ∥∂xΦ∥s′ ≤ Cδ,n,s,s′ ∥Φ∥s for Φ ∈ A+,n_δ(Γ).Source: §7.2.2 · p. 186

7.2.3 The general case

• For every Θ ∈ (0, 2π), the formal-series results extend beyond rational angles, and Φ^0_{d,n} converges for every (d, n).The extension uses an angle-dependent increasing family of thresholds sδ,n.Source: §7.2.3 · p. 188
• The general-case profile series has a C1 limit with uniform convergence on bounded regions and a stated remainder estimate.The passages identify the limit and estimate, but the estimate’s full formula is not available in the excerpts.Source: §7.2.3 · pp. 191–192

7.3 Proof of the modal decomposition

• Section 7.3 proves Theorems 7.1.1 and 7.1.3, identified as Chapter 7’s main results.Source: §7.3 · p. 193

7.3.1 Proof of Theorem 7.1.1

• The corner modal decomposition is classical, and its key lemma bounds coefficients s_d by C R^-d for each 0 < R < R_s.The ensuing series and its gradient converge locally uniformly.Source: §7.3.1 · pp. 194–195
• The proof establishes the stated estimates and uniform convergence of the modal decomposition on each Ω∩B(0, R), for R∈(0, Rs).Source: §7.3.1 · p. 197
• The decomposition’s convergence domain is restricted by the behavior of the coefficients σd(up−n,ℓ) as d→∞.Source: §7.3.1 · p. 197

7.3.2 Proof of Theorem 7.1.3 and Corollary 7.1.4

• The truncated modal decomposition T<q(u0p,ℓ) converges uniformly to up,ℓ on each Ω∩B(0, R), for R∈(0, Rs).Source: §7.3.2 · p. 199

8.1 Complements at infinity: a numerical method to compute the corner profiles

• The proposed method computes corner profiles and their asymptotic coefficients, with convergence estimates and numerical results to validate the theoretical results.The profiles’ behavior at infinity makes their coefficients difficult to extract on a truncated domain, motivating a dedicated numerical method.Source: §8.1 · pp. 200–201

8.1.1 Model problem and state of the art

• Complements at infinity enrich the Galerkin space with known functions near infinity, with their coefficients left unknown, to capture the solution’s slow decay.Unlike singular-function methods that enrich a space near a corner, this approach can use as many complements at infinity as needed for a high-order method.Source: §8.1.1 · p. 203

8.1.2 Presentation of the numerical method

• The variational method approximates the solution using a truncated-domain component and complements at infinity, and recovers asymptotic coefficients from the complement coefficients.The approximate problem is well posed by the Lax–Milgram theorem; the total approximation is χSA + uL.Source: §8.1.2 · p. 204

8.1.3 Proofs of convergence

• The method’s convergence and error estimates hold as L tends to infinity at fixed N; convergence as N tends to infinity at fixed L is not established.The authors report that increasing N yields faster convergence, while numerical tests suggest quadrature error is close to machine error.Source: §8.1.3 · p. 205
• The coefficient-estimator errors tend to zero as L grows, with faster convergence for small n and slower convergence near N; taking N above the number of desired coefficients improves their precision.Numerical slopes are steeper in absolute value than the theorem predicts, suggesting its convergence-rate estimate is not optimal; measured coefficient errors also generally decrease as N increases and increase with n.Source: §8.1.3 · p. 206, §8.1.4 · p. 211

8.1.4 Numerical validation

• With N = 4 and L = 4, the maximal solution error is 8 · 10^-5, much smaller than with the homogeneous Dirichlet truncation; corner irregularity limits the numerical setup.The exact corner profiles have explicit expressions, but one candidate is insufficiently regular at the corner for the high-order Lagrange finite elements used to avoid instability.Source: §8.1.4 · pp. 208–209
• For N = 4, the coefficient is computed with very high precision: its relative error is 2 · 10^-9 at L = 5, falling to 4 · 10^-11 at L = 12.5 and h = 0.035.Source: §8.1.4 · p. 209

8.2 Numerical computation of the far fields

• The numerical far-field computation targets validation of the asymptotic expansion and its error estimate, using inductively constructed fields with finitely many nonzero terms for each order.The method must address corner singularities and non-smooth finite-element derivatives in boundary data; it also truncates the unbounded domain with a PML.Source: §8.2 · p. 213, §8.2.1 · p. 213

8.2.1 Presentation of the numerical method

• A DtN map on an artificial boundary around the corner handles corner-divergent far fields through modal decomposition, while the method also computes their asymptotic coefficients.The fields’ corner behavior makes them unsuitable as ordinary variational solutions, and their heterogeneous Dirichlet data involve derivatives of previously computed fields.Source: §8.2.1 · pp. 213–214
• The truncated-series DtN condition computes far fields while accommodating their corner divergence, approximating them near the corner, and providing their corner coefficients.The condition is approximated by truncating the series defining F.Source: §8.2.1 · p. 215
• The finite-element procedure regularizes derivative computations to reduce high-frequency oscillations; the factor 4 in the inductive computation was reported to work best in practice.The regularization parameter scales with the square of the mesh step by dimensional analysis.Source: §8.2.1 · p. 216
• Far-field computations validate the error estimate but accumulate numerical errors; the order-3 and order-5 effective models reduce cost and errors and use easier Robin and Ventcel conditions.The respective effective-model boundary conditions are Robin for order 3 and Ventcel for order 5.Source: §8.2.1 · pp. 216–217

8.2.2 Numerical results

• The tests use angles Θ = 2.5, π, and 4, a rounded layer, and thicknesses ε ∈ {0.04, 0.065, 0.1, 0.15, 0.25}, with finely meshed reference solutions.P1 Lagrange finite elements were used because higher-order elements appeared to handle the derivative computation poorly.Source: §8.2.2 · p. 217
• For Θ = 4 and Θ = π, remainders decrease as p increases; the far fields qualitatively match the remainders, with log terms contributing when Θ = π.For Θ = 4, the expansion has no ln ε; for Θ = π, the remainder is especially close to u_p,0 in practice.Source: §8.2.2 · p. 218, Table 8.1 · p. 219, Table 8.2 · p. 220
• The relative-error curves show the expected convergence order when regressions use the initial points, despite plateaux attributed to numerical errors; the expansion is verified through order 5.The curves use the H1 error normalized by the H1 norm of u0,0 on the computational domain, excluding the PML.Source: §8.2.2 · pp. 217–218, Figure 8.9 · p. 221, Figure 8.10 · p. 221

9.1 Introduction

• The third- and fifth-order models compute an approximation directly rather than summing many far fields, reducing computational cost, while leaving corner-profile computation unchanged.The corner profiles are independent of frequency and source term and can therefore be computed once in advance.Source: §9.1 · p. 223

9.1.1 Description of the models

• Away from the corner, the models impose effective Robin or Ventcel conditions; near it, a finite-rank DtN condition on an artificial boundary incorporates corner singularities and matching coefficients.The DtN map is built from an approximate modal decomposition and matches radial derivatives for the first N modes.Source: §9.1.1 · pp. 224, 226

9.1.2 Statement of the main results

• The models are well-posed and uniformly stable when ε is sufficiently small; without the stability condition, they remain well-posed except at a discrete set of ε values.The stability argument shows the DtN contribution is O(ε)-close to a positive operator, yielding uniform coercivity for sufficiently small ε.Source: §9.1.2 · pp. 226–227
• The first term of (9.1.8) is O(ε^m), weighted by 1/(R°)^m; the second, from DtN truncation, tends to zero geometrically with N, making O(ε^m) dominant.The first term also has an N^3 factor for technical reasons, improvable to N^(2.5+η) for any η > 0 with a changed constant.Source: §9.1.2 · p. 227

9.2.1 Preliminary decomposition

• The approximate modal decomposition of uε is constructed to order ε^m from the known modal decomposition of the far fields, providing the basis for the DtN operators.The passage identifies the far-field modes as the starting point for the construction.Source: §9.2.1 · p. 228
• The preliminary modes account for the layer and approximately satisfy the effective boundary condition, but omit corner matching information needed to represent the perturbation and matching coefficients.The omitted information includes algebraic matching coefficients and corner-profile coefficients.Source: §9.2.1 · p. 229, §9.2.2 · p. 232
• Proposition 9.2.3 gives an approximate modal decomposition of u_ε with error O(ε^m ln^ℓ_m ε), and thus approximates the exact solution at that order.The estimate applies for r < R when ε ≤ Mr.Source: §9.2.1 · p. 230

9.2.2 Reexpression of the far fields’ corner conditions

• Proposition 9.2.5 reexpresses negative-index corner coefficients through positive-index coefficients, incorporating the corner constraints into a second modal decomposition.The new Ψ modes depend on the coefficients cu←u, while the preliminary modes omit matching conditions around the corner.Source: §9.2.2 · pp. 232–233
• An induction on the expansion order reexpresses far-field corner conditions by applying the induction hypothesis to nonzero convolution terms.The argument adds the first term of the expression after handling the terms whose indices satisfy p′ < p.Source: §9.2.2 · p. 234

9.2.3 Final decomposition

• The final modal decomposition approximates the field with truncation errors quantified at order ε^m ln^ℓ_m ε.The estimate applies for r and θ satisfying ε⟨ln r⟩^3 ≤ Mr; the proof uses the simplifying assumption N ≥ mΘ/π, which can be avoided with an estimate for missing modes.Source: §9.2.3 · pp. 237–238

9.3.1 Definition of the DtN operators

• The DtN operators are defined from linear forms on the span of the modes, using the inverse of their Gram matrix.The matrix is invertible near ε = 0 and outside a discrete set of ε values, so the operators are well-defined there.Source: §9.3.1 · pp. 239–240

9.3.2 Explicit description of the DtN modes except for particular angles

• For angles outside the exceptional sets E_m, the far-field matching conditions simplify at orders p < 5, yielding simpler DtN modes.These simplifications follow from the matching coefficients having simple expressions in those cases.Source: §9.3.2 · p. 240
• For small angles, Proposition 9.3.3 gives explicit expressions for the DtN modes, while the proof handles irrational and rational angle cases separately and treats (Θ = π/3, m = 5) as exceptional.The listed mode expressions distinguish the order m and angle conditions.Source: §9.3.2 · p. 241

9.4 Well-posedness of the effective models

• When ℑ(ω) ≠ 0, uniform coercivity for sufficiently small ε yields well-posedness and uniform stability of the order-3 and order-5 effective models.The coercivity argument uses the fact that the relevant sesquilinear forms can be expressed using real nonnegative terms and the sign of ℑ(ω).Source: §9.4 · p. 242

9.4.1 Asymptotic behavior of the DtN operators for a small ε

• The DtN operator S_N,0 contributes positively, and the difference between the ε-dependent and ε=0 DtN operators is O(ε) in L(L2(Σ◦)) as ε → 0.The estimates also provide bounds under the condition ε⟨ln R◦⟩³ ≤ R◦, including bounds in a scaled H2 norm used for error estimates.Source: §9.4.1 · pp. 242, 245
• The modal decomposition is used to analyze the DtN operators as ε tends to zero, with sine orthogonality supporting their good properties and uniform stability.The text notes that the alternative decomposition would likely fail to converge as ε → 0, preventing a proof of uniform stability.Source: §9.4.1 · p. 247

9.4.2 Proof of well-posedness and stability

• Under the stated stability condition, the effective model has a unique solution and is uniformly stable for both m = 3 and m = 5.The proof establishes coercivity and applies Lax–Milgram’s theorem.Source: §9.4.2 · pp. 248–249

9.5 Convergence of the effective models

• The initial convergence estimate is O(ε^(m−1)) plus an N-dependent truncation error; later consistency estimates are used to improve it to O(ε^m).The preliminary order is lower because the effective boundary-condition error is only order m − 1.Source: §9.5 · p. 250, §9.5.1 · p. 251

9.5.1 Preliminary error estimates of order 2 and 4

• Preliminary estimates control the error at order O(ε^(m−1)) before consistency estimates yield the target order O(ε^m).The error analysis also bounds the DtN operator and accounts for the stability condition.Source: §9.5.1 · p. 251

9.5.2 Tools of elliptic regularity

• A local elliptic-regularity result bounds the boundary trace by O(ε), and a diffeomorphism and symmetry extension adapt it to the corner at P◦.The corner has angle π/2, so the geometry is deformed locally into a quarter plane before extension to a half plane.Source: §9.5.2 · pp. 254–255
• A corner-adapted C1 diffeomorphism is built to be Cartesian near Γ◦ and polar near Σ◦, preserving Neumann, Robin, or Ventcel boundary conditions for elliptic regularity arguments.The construction transitions between two coordinate maps with a partition of unity and requires proving differentiability at the corner and Lipschitz continuity of its derivative.Source: §9.5.2 · p. 255

9.5.3 Final error estimates of order 3 and 5

• Under the stated stability condition, the effective-model problem has a unique solution and admits final error estimates for orders 3 and 5.The argument uses uniform coercivity and continuity, a boundary-data lift, and Lax–Milgram; the supplied passages do not expose the final estimate formulas.Source: §9.5.3 · pp. 256–258

9.6 Numerical results

• Effective Robin and Ventcel conditions are easier to discretize than far-field conditions, which require high-order derivatives, while the order-5 DtN term is non-classical.The order-5 condition couples a boundary integral of the unknown with evaluation of a test function at the corner.Source: §9.6 · pp. 258–259
• The Gram matrix can be severely ill-conditioned, with the ratio of its largest to smallest diagonal coefficient reaching about 10^15; a preconditioner is used for inversion.The authors attribute the conditioning to geometric behavior of the diagonal coefficients as mode index increases.Source: §9.6 · p. 259
• Errors for models of orders 1 and 3 converge at their expected rates, while order-5 accuracy depends on angle and DtN parameter choices.The numerical experiments use 4–12 modes; increasing the mode count can worsen error, consistent with the theorem’s increasing N^3 factor.Source: §9.6 · pp. 259–260
• Order-5 errors converge with slopes 4.7–4.8 for Θ=π and 4, but not Θ=2.5; the models validate in most cases, with minimum relative errors around 10^-4.For Θ=2.5, the order-5 error is concentrated near the DtN boundary; the authors link the lack of observed fifth-order convergence tentatively to the choice of DtN parameters.Source: §9.6 · p. 260, Figure 9.3 · p. 264, Figure 9.4 · p. 264

Proofs of Lemmas 9.5.6–9.5.8

• Finite differences establish local H2 regularity; the boundary condition then yields an H1/2 trace estimate of order O(ε).The Ventcel case requires an additional boundary regularity step because the tangential second derivative is not initially controlled.Source: Appendix: proofs of the technical… · pp. 266–267
• For the Ventcel condition, Fourier transformation of the boundary relation provides control of the localized trace after the extra regularity step.The Robin trace estimate follows more directly from its boundary condition and the local H2 bound.Source: Appendix: proofs of the technical… · p. 267
• A coordinate change preserves the relevant corner geometry and has an invertible derivative at the corner, enabling the local argument to be transferred.The construction gives Dφ(0) = Diag(1, R◦).Source: Appendix: proofs of the technical… · pp. 267–268
• Rescaling to unit corner radius and applying the local estimate gives an H1/2 boundary bound proportional to ε on the target interval.The proof rescales ε together with the geometry before transferring the estimate back.Source: Appendix: proofs of the technical… · pp. 268–269

Stability and error estimates

• The appendix indicates an improved error estimate and modifies intermediate results to make the number-of-modes dependence more precise.The refinements increase the technicality of the proofs in Chapters 7 and 9.Source: Appendix: behavior of the stability… · p. 270
• The revised norm estimates increase the norm index rather than the scale parameter, which the appendix identifies as sharper.The appendix also replaces a geometric bound on ⟨d⟩ with a more precise estimate.Source: Appendix: behavior of the stability… · p. 270
• The stated stability conditions depend on the model order and mode count, with separate forms for m = 3 and m = 5.The appendix gives revised thresholds for both model orders.Source: Appendix: behavior of the stability… · p. 272

Stability and error estimate refinements

• Under the stability condition, the relevant coefficient term is controlled to prove that the effective models are uniformly coercive.The proof follows the coercivity strategy used for the earlier well-posedness result.Source: R /N 2, CDtN,2 · p. 272
• The proof changes the stability constants and related lemmas to use a constant proportional to 1/N^3.The appendix says the well-posedness theorem remains unchanged after these proof modifications.Source: R /N 2, CDtN,2 · p. 272
• For the m = 5 argument, the appendix replaces N^3 by N^(2.5+η) in the remaining proof steps.This modification follows the generalized estimate described for Proposition 9.4.8.Source: R /N 2, CDtN,2 · pp. 272–273

Conclusion and perspectives

• An explicit corner-singularity basis, operators, and formal series support generalizable matching calculations for the arbitrary-order expansion.The symbolic module computes matching coefficients and radial asymptotic behaviors exactly and efficiently.Source: Conclusion and perspectives · p. 274
• The expansion simplifies at every order for irrational π/Θ and at the first orders for rational π/Θ, but the general form remains needed at Θ = π.The stated simplifications include simpler matching conditions and no powers of ln ε.Source: Conclusion and perspectives · p. 274
• An enriched Galerkin method with complements at infinity computes corner profiles, and numerical results validate the expansion through order 5 for several angles.The complements at infinity are used in the numerical method for computing the profiles.Source: Conclusion and perspectives · p. 274
• The order-3 and order-5 effective models use a near-corner DtN condition; the thesis proves well-posedness, uniform stability, and error estimates, and provides numerical results.The DtN construction uses a modal decomposition that accounts for singular far-field behavior.Source: Conclusion and perspectives · p. 274
• The angle range for which naive models converge remains undetermined; the authors conjecture improved far-from-corner rates and note the simpler implementation of those models.The conjectured rates are tied to the first far fields with non-trivial corner conditions.Source: Conclusion and perspectives · pp. 274–275
• A related DtN condition for computing corner profiles could make the finite-element matrix sparser, but its error analysis may be more complex and coefficient-error bounds are unknown.The proposed alternative would replace complements at infinity.Source: Conclusion and perspectives · p. 275
• Future extensions include Neumann corners, where the expansion may involve powers of ln ε, and three-dimensional geometries requiring suitable corner-singularity bases.The work has not yet investigated adapting the other results to Neumann conditions; conical and polyhedral points are identified as harder cases.Source: Conclusion and perspectives · p. 275
• Finite periodic layers remain a major open direction because the matching between the corner and layer zones is unresolved.The proposed approach also requires computing layer profiles in addition to corner layers.Source: Conclusion and perspectives · p. 275