Optimized domain decomposition methods for the spherical Laplacian

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Article

Reference

Optimized domain decomposition methods for the spherical Laplacian

LOISEL, Sébastien, et al.

LOISEL, Sébastien, et al . Optimized domain decomposition methods for the spherical Laplacian. SIAM Journal on Numerical Analysis , 2010, vol. 48, no. 2, p. 524–551

DOI : 10.1137/080727014

Available at:

http://archive-ouverte.unige.ch/unige:6259

Disclaimer: layout of this document may differ from the published version.

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Laplaian

S. Loisel

∗

J. Cté

†

M. J.Gander

∗

L.Laayouni

‡

A. Qaddouri

†

September 21, 2009

Abstrat

The Shwarz iteration deomposes a boundary value problem overa large domain

Ω

^into ^smaller

subproblems by iteratively solving Dirihlet problems on a over

Ω ¹ , ..., Ω 𝑝

^of

Ω

^. ^In ^this ^paper, ^we

disussalternatetransmissiononditionsthatleadtofasteronvergenefortheLaplaianonthesphere

Ω

^. ^We ^look ât ^Robin ônditions, ^seond ôrder ^tangential ônditions ând â ^disretized ^version ôf ân

optimalbutnonloaloperator.

1 Introdution

AttheheartofnumerialweatherpreditionalgorithmslieaLaplaeandpositivedeniteHelmholtzprob-

lemsonthesphere[35℄. Reently,therehasbeeninterestinusingniteelements[5℄anddomaindeomposi-

tionmethods[4,26℄. TheShwarziteration[23,24,25℄anditsoptimizedvariantshavebeenverysuessful

andthesubjetofmuhreentresearh. WenowoutlinethehistoryofOptimizedShwarzMethods(OSM)

andreferto[14℄forfurtherdetails,aswellasaompletederivationforHelmholtzandLaplaeproblems in

theplane.

The optimizedShwarzmethod was introduedin [2,31, 32℄under various names. SinetheOSMhas

oneormorefreeparameters,whihneedtobehosenarefullytoguaranteethebestonvergenerate,there

followedaneorttondthebestpossibleparameterhoies[13,21,20℄. Despitethesepositivedevelopments,

aproofofonvergeneforageneralsituationhasprovenelusive[22,27℄. Onewayofobtainingonvergene

istodenearelaxationofthemethod[7℄.

TheusualmethodforoptimizingthefreeparametersistotakeaFouriertransformofthepartialdier-

entialequation, obtaininganexpliitreurrenerelationfortheiteration. This works onlyinspeialases

(e.g., thedomain is aretangle and the dierentialoperator is the Laplaianwith homogeneousDirihlet

onditions). In addition to theLaplae and Helmholtzproblems [19℄, the method an beused for various

otheranonialproblems(f. [9,18,17,29,28℄foronvetion-diusionproblems,[16℄forthewaveequation,

[8℄ for uid dynamis, [33℄ for the shallow water equation). The urrent paper is a ontinuation of this

researh.

OurpaperprovidesanOSMforthespherialLaplaian,adierentialoperatorwhihhadpreviouslynot

beenanalyzed,usingFourieranalysis. However,webrieymentionthattherehasbeenmuhreentresearh

onvariousother aspetsoftheOSM.Forinstane, domainswithornersan giverisetosingularities,and

this isanalyzed in [3℄. Inasimilar vein, dierentialoperators with disontinuousoeientsoften leadto

ill-onditionedproblems, and anOSMforthis situation isprovidedin [12℄. A ompletelydierent issueis

to phrasethe OSMin analgebrai way. In atypialappliation, one hasaode to omputethe stiness

matrix

𝐴

^,^butône^wouldîdeally^prefer^not^to^have^to^rewrite^theêntireôdeⁱⁿôrder^toîmplementânÔSM.

Inthatvein,[34℄providesaframeworkforexpressingtheOSMin thelanguageofmatries.

∗

Setion de Mathématiques, Université de Genéve, CP 64, CH-1211 Geneva, Switzerland (sloiselgmail.om,

gandermath.unige.h).

†

Reherheenprévisionnumérique,EnvironnementCanada,2121Route Transanadienne,Dorval, Québe,Canada H9P

1J3(jean.otee.g.a, abdessamad.qaddourie. g. a)

‡

Shool of Siene and Engineering Al AkhawaynUniversity Avenue Hassan II, 53000 P.O. Box 2165, Ifrane, Moroo

(L.Laayouniaui.ma)

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deompositionmethods areoftennaturallyparallel,andheneanbeusedonlargelustersandsuperom-

puters. Thealgorithmthat runsoneahproessorisasmall,loalversionoftheglobalproblem,soproven

sequentialodesan beadapted and beome loal solverswhih will thenrun in parallel onsuh lusters.

Insuhasenario,animportantquestionisthat ofsaling.

In the simplest saling senario, one xes the number

𝑝

^of subdomains, and one piks subdomains

Ω 1 , . . . , Ω 𝑝

ⁱⁿ^suhâ^way^that^theôverlap^betweenâdjaent^subdomainsîsêxatlyôneêlement^thik. În^this

situation,itiswellknown[36℄thatthe(1-level)Shwarzmethodhasaonvergenefatorof

𝜌 𝐶𝑆 = 1 − 𝑂(ℎ)

^.

Inotherwords,ateahstepoftheiterativemethod,theerrorismultipliedbytheoeient

𝜌 𝐶𝑆 < 1

^. ^Beause

this oeientdepends on

ℎ

^, ^and ^tends ^to ¹^when

ℎ

^tends^to ^zero,^we^see ^that ^more^and ^more^iterations

willbeneededtoreahagiventoleraneaswemake

ℎ

^smaller. Ît îs^preferable^to ^ndânâlgorithm^whose

onvergenefatoreitherdoesnotapproah

1

^,ôrât^leastâpproahes

1

^not^too^quikly^when

ℎ

^tends^to^zero.

In this ontext, themain advantageof theOptimized Shwarzmethods is that they havemuh better

onvergenefatorsthanthelassialShwarzmethod. Forexample,theone-sidedoverlappingOO0method

hasaonvergenefatorof

𝜌 𝑂𝑂0 = 1 − 𝑂(ℎ ^1/3 )

^,^and^theoverlappingOO2methodhasaonvergenefator of

𝜌 𝑂𝑂2 = 1 − 𝑂(ℎ ^1/5 )

^.

Inadditiontooverlappingmethods,Lions[25℄proposedtouseRobintransmissiononditionsalongthe

interfae to obtain nonoverlappingdomain deomposition methods. There is no nonoverlapping lassial

Shwarzmethodwithwhihtoompare,buttheOO0andOO2methodsanbeusedinthenonoverlapping

ontext. Theonvergenefators are

1 − 𝑂(ℎ ^1/2 )

^and

1 − 𝑂(ℎ ^1/4 )

^, ^for^thenonoverlappingOO0 andOO2 methods,respetively.

Although ouranalysis islimitedtolatitudinalsubdomains,wewillseein Setion5thatweobtainvery

good performane,evenwhensubdomainshavemorevariedshapes.

Wenotethat,ifthenumberofsubdomains

𝑝

^inreases^as

ℎ

^tends^to^zero,îtîs^neessary^to^designâ^2-level

algorithm to maintaingood performane. Ina 2-levelalgorithm, detailed informationsfrom neighbors are

ombinedwithlesspreise informationstoredonalow-resolutionoarsegrid. Thedesignand analysisof

anOptimizedShwarzMethodwithaoarsegridorretionisahallengingproblemwhihwillbeanalyzed

inanupomingpaper[10℄.

1.1 Our ontributions

Inthispaper,weintrodueimprovedtransmissionoperators fortheLaplaeproblemonthe unitsphere

Ω

in

ℝ ³

^. ^We ^modify ^the ^Shwarz îteration ^by ^replaing ^the^Dirihlet ônditions ôn^the înterfaes ^by ^Robin

or seondordertangentialonditions. Thisallowsusto signiantlyimprovetheonvergenefatorofthe

iteration. Toanalyzetheonvergenerateandoptimizetheoeients,weusetheFouriertransformonthe

sphere. Inadditionto theusual Robinand seondordertangentialonditions,wealsodisusstheuseofa

nonloaloperatoralongtheinterfaes. ThisnonloaloperatorisrelatedtothesquarerootoftheLaplaian

oftheinterfaesandwasrstintroduedfornumerialalulationsin[4,26℄. Whiletheontinuousanalysis

showsthat theseoperators resultin aniterationthatonvergesin twosteps,in pratiewedonotseethis

exatbehavior. However,theresultingiterationseemstoonvergeataveryfastratethatisindependentof

themeshsize

ℎ

ôr^the^thiknessôf^theôverlap

𝐿

^.

Wehighlightfour ontributionsofthispaper. First,wehaveaninnovativeuseoftheenvelopetheorem

toestablishanequiosillationproperty. Seond,wehighlightourontinuousanddisreteanalysesandtheir

omparison. Third,ouranalysistakesplaeonthesphere,whihisidealforweatherandlimatesimulation.

Fourth, we provide a nite element method whih is appropriate for spherial disretizations. Sine the

spherialLaplaianissingular,thehoieoftheniteelementspaeisimportantandapoorhoieleadsto

divergentiterationsandpoorlyonditionedsystems. Weshowthatwiththeproperhoieofbasisfuntions,

theonvergenefatoroftheoptimizedShwarzmethodsareunaeted.

This paperis organizedas follows. InSetion 2, weintrodue ourmodel problem andstate our main

results. InSetion 3, we review the Laplae operator on the sphereand reall theShwarz iteration and

its onvergene estimates, previously published in [4℄; we also give a new semidisrete estimate whih is

substantiallysimilartotheontinuousone. Insetion4,weperformtheFourieranalysisofthenewOptimized

ShwarzMethods. Insetion5,wepresentnumerialresultsthat agreewiththetheoretialpreditions.

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Ω ₂

Ω ₁

a b

Ω ₂

Ω 1

a

Ω ₃

2 b ₂

a 3 b 1

Figure1: Latitudinal domaindeomposition. Left: twosubdomains;right: multiplesubdomains.

2 OSM on the sphere

Considerthe followingiteration. Let

𝑏 < 𝑎

^. ^Begin ^with ^random ^andidate ^solutions

𝑢 0

^and

𝑣 0

^. ^Dene

𝑢 𝑘+1

^and

𝑣 𝑘+1

iterativelyby:

⎧

 

⎨

 

⎩

Δ𝑢 𝑘+1 = 𝑓

ⁱⁿ

Ω 1 = { (𝜑, 𝜃) ∣ 0 ≤ 𝜑 < 𝑎 } , 𝑢 𝑘+1 (𝑎, 𝜃) = 𝑣 𝑘 (𝑎, 𝜃) 𝜃 ∈ [0, 2𝜋),

Δ𝑣 𝑘+1 = 𝑓

ⁱⁿ

Ω 2 = { (𝜑, 𝜃) ∣ 𝑏 < 𝜑 ≤ 𝜋 } , 𝑣 𝑘+1 (𝑏, 𝜃) = 𝑢 𝑘 (𝑏, 𝜃) 𝜃 ∈ [0, 2𝜋);

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(see gure 1.) This is the Shwarz iteration for two latitudinal subdomains. The idea of the Optimized

Shwarz Methods is to replae the Dirihlet onditions on the interfae by Robin or other transmission

onditions. Themodiediterationis:

⎧

 

⎨

 

⎩

Δ𝑢 𝑘+1 = 𝑓

ⁱⁿ

Ω 1

((𝜓 + _∂𝜑 ^∂ )𝑢 𝑘+1 )(𝑎, 𝜃) = ((𝜓 + _∂𝜑 ^∂ )𝑣 𝑘 )(𝑎, 𝜃) 𝜃 ∈ [0, 2𝜋),

Δ𝑣 𝑘+1 = 𝑓

ⁱⁿ

Ω 2

((𝜉 + _∂𝜑 ^∂ )𝑣 𝑘+1 )(𝑏, 𝜃) = ((𝜉 + _∂𝜑 ^∂ )𝑢 𝑘 )(𝑏, 𝜃) 𝜃 ∈ [0, 2𝜋);

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where

𝜓

^and

𝜉

âre êither ^real ôeients ôr ^trae ôperators, ând

Ω 1 , Ω 2

âre âs ^previously ^dened. Îf

𝜓

and

𝜉

âre^dierential^traeôperatorsôfôrder

𝑘

^,^we^say^that^theôptimized^Shwarz^methodîsÔptimizedôf

Order

𝑘

^, ôrÔOk^[15℄. ^Choiesînlude:

1.

(𝜓𝑤)(𝜃) = 𝑐𝑤(𝜃)

^where

𝑐

îsâ^realôeient. ^This^resultsⁱⁿ â^Robinôr ÔO0transmissionondition.

2.

(𝜓𝑤)(𝜃) = 𝑐𝑤(𝜃)+𝑑𝑤 ^′′ (𝜃)

^,^where

𝑐

^and

𝑑

âre^realôeients. ^This^resultsⁱⁿâ^seondôrdertangential, or OO2transmissionondition.

3. Anonloalhoieof

𝜓

^leading^toânîteration^thatônvergesⁱⁿ^two^steps.

Wewill showin Setion 3thattheonvergenefator depends onthethiknessof theoverlap

𝐿

^, ^given

by

𝐿 := − 2 log 1 + cos 𝑎 sin 𝑎

sin 𝑏 1 + cos 𝑏 ≥ 0.

OurmainresultforRobin(orOO0)transmissiononditionsisas follows.

Theorem2.1(OptimizedRobinorOO0oeients). Considerthe iteration(2)with

(𝜓𝑤)(𝜃) = 𝑐𝑤(𝜃)

^and

(𝜉𝑤)(𝜃) = 𝑑𝑤(𝜃)

^. ^Set

𝑥 = − 𝑑 sin 𝑏

^,

𝑦 = 𝑐 sin 𝑎

^. ^Let

ℎ = 1/𝑁

^be^the ^grid ^size.

∙

^The ône-sidedônditionîs

𝑥 = 𝑦

^. ^With^thisôndition,ând^furtherâssuming^that

𝐿 < 4/3

^and

𝐿 > ℎ ^𝛼

for some

𝛼 < 3/2

^, ^there îs â ûnique ôptimized ^hoie ôf

𝑥

^leading ^to ^the ^best ^possible ^onvergene

fator. Using

𝑥 = 𝑦 = ˆ 𝑥 1 = 𝐿 ⁻ ¹ ³

^, ^theônvergene^fatorêvery ôther ^stepîs

𝜌 = 1 − 4𝐿 ¹ ³ + 𝑂(𝐿 ² ³ )

^.

∙

^If

𝑥 ∕ = 𝑦

îs âllowed, ^the ôptimized^hoie

(𝑐, 𝑑)

^is ^two-sided. ^Assume^that

𝐿 > ℎ ^𝛼

^,

𝛼 < 5/4

^. ^There^is

a unique

(𝑐, 𝑑)

^leading ^to^the ^best^onvergene^fator. ^Using

𝑥 = 8 ⁻ ¹ ⁵ 𝐿 ⁻ ¹ ⁵

^and

𝑦 = ⁸

2 5

2 𝐿 ⁻ ³ ⁵

^leads ^to^a

onvergenefator of

𝜌 = 1 − 2 ⋅ 2 ³ ⁵ 𝐿 ¹ ⁵ + 𝑂(𝐿 ² ⁵ )

^every^other ^step.

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Theonvergenefator

𝜌 < 1

^multiplies^the

𝐿 ²

êrrorôf^the^traeônêah^subdomainêveryôther^step. ^We

seethatusingasymmetri Robinonditionsleadstoanimprovementonthenumberofiterationsrequired

for a given tolerane, from

𝑂(𝐿 ⁻ ¹ ³ )

^to

𝑂(𝐿 ⁻ ¹ ⁵ )

^. ^The ône-sided êstimate ^however îs êasier ^to ûse ^with

𝑝

subdomains,andsowewouldbeinterestedinahievinga

𝑂(𝐿 ⁻ ¹ ⁵ )

âlgorithmûsingônlyône-sidedêstimates.

Thisispossibleifweuseseondordertangentialtransmissiononditions.

Theorem 2.2 (One-sidedseond order tangential,or OO2,optimized oeients). Consider the iteration

(2)with

(𝜓𝑤)(𝜃) = 𝑐𝑤(𝜃) + 𝑑𝑤 ^′′ (𝜃)

^,

(𝜉𝑤)(𝜃) = ˜ 𝑐𝑤(𝜃) + ˜ 𝑑𝑤 ^′′ (𝜃)

^. ^Let

𝑠 = 𝑐 sin 𝑎 = − 𝑐 ˜ sin 𝑏

^and

𝑡 = 𝑑 sin 𝑎 =

− 𝑑 ˜ sin 𝑏

^, ^whih ^is ^the ^one-sided assumption. Assumethat

𝐿 < ℎ ^𝛼

^,

𝛼 < 5/4

^. ^There îsâ ûnique^hoie ôf

𝑠, 𝑡

^leâding^to^the^best^possibleônvergene^fator. Ûsing

𝑠 = ¹ ₂ 2 ³ ⁵ 𝐿 ⁻ ¹ ⁵

^and

𝑡 = ¹ ₂ 2 ¹ ⁵ 𝐿 ³ ⁵

^leads^to^a^onvergene

fatorof

𝜌 = 1 − 4 ⋅ 2 ² ⁵ 𝐿 ¹ ⁵ + 𝑂(𝐿 ² ⁵ )

^every^other ^step.

Theonvergerateestimateshold evenif

𝐿

^is^smaller^than

ℎ ^𝛼

^(e.g., ^when^the^grid^is^highlyanisotropi, with

𝛼 = 3/2

^or

5/4

^,âsâbove),^but^then^theôeients^given^may^no^longer^beôptimized. Ânêxtremeâse

iswhen

𝐿 = 0

^,^anonoverlappingase,andweanalyzethisaseinSetion3.1. Themain aseis

𝐿 = 𝑂(ℎ)

andishandledbytheTheorems2.1and2.2.

Our numerialexamplesarebased ontwodierentodes. Many odesin meteorologyuse thedisrete

Fouriertransformin thelongitudevariable

𝜃

^to^obtain^asemispetral solver. This solvermakestheimple- mentationofourthreetypesoftransmissiononditionsequallyeasy,forlatitudinalinterfaes. However,suh

solversdonoteasilyhandlearbitrarilyshapedsubdomains,andthegridshavesingularitiesat thepoles.

Ourseond odeisaniteelementsolveronthesphere

Ω

^. ^Thevariationalformulationofthespherial Laplaianleadsto integralsonthesphere

Ω

^,^and ^so^we^must ^hoose ^pieewise

𝐶 ¹

^basis ^funtions^for ^some

niteelementspaeonthesphere

Ω

^. ^Weûseâ^standard^projetion^tehnique^to^generate^spherialêlements

frompieewiselinearelementsonapolyhedralapproximationof thesphere

Ω

^.

3 The Laplae operator on the sphere

Ω

WetaketheLaplaeoperatorin

ℝ ³

^,^given^by

Δ𝑢 = 𝑢 𝑥𝑥 + 𝑢 𝑦𝑦 + 𝑢 𝑧𝑧 ,

rephraseitinspherialoordinatesandset

∂𝑢

∂𝑟 = 0

^to ^obtain

Δ𝑢 = 1

sin ² 𝜑

∂ ² 𝑢

∂𝜃 ² + 1 sin 𝜑

∂

∂𝜑 (

sin 𝜑 ∂𝑢

∂𝜑 )

,

where

𝜑 ∈ [0, 𝜋]

îs^theôlatitudeând

𝜃 ∈ [ − 𝜋, 𝜋]

^the^longitude.

3.1 The solution of the Laplae problem

WetakeaFouriertransformin

𝜃

^but^notⁱⁿ

𝜑

^;^this ^lets^us^analyze^domaindeompositionswithlatitudinal boundaries. TheLaplaianbeomes

Δˆ ˆ 𝑢(𝜑, 𝑚) = − 𝑚 ²

sin ² 𝜑 𝑢(𝜑, 𝑚) + ˆ 1 sin 𝜑

∂

∂𝜑 (

sin 𝜑 ∂ 𝑢(𝜑, 𝑚) ˆ

∂𝜑 )

, 𝜑 ∈ [0, 𝜋], 𝑚 ∈ ℤ.

⁽³⁾

Forboundaryonditions,theperiodiityin

𝜃

îs^takenâreôf^by^the^Fôurierdeomposition.Thepolesimpose that

𝑢(0, 𝜃)

^and

𝑢(𝜋, 𝜃)

^do^not^varyⁱⁿ

𝜃

^. ^F^or

𝑚 ∕ = 0

^this^is^equivalent^to

ˆ

𝑢(0, 𝑚) = 𝑢(𝜋, 𝑚) = 0, 𝑚 ˆ ∈ ℤ, 𝑚 ∕ = 0.

⁽⁴⁾

For

𝑚 = 0

^,^the^relation

𝑢 𝜑 (0, 𝜃) = − 𝑢 𝜑 (0, 𝜃 + 𝜋)

^leads^to

∫ 2𝜋

0 𝑢 𝜑 (0, 𝜃) 𝑑𝜃 = − ∫ 2𝜋

0 𝑢 𝜑 (0, 𝜃) 𝑑𝜃

^,^i.e.,

ˆ

𝑢 𝜑 (0, 0) = 𝑢 ˆ 𝜑 (𝜋, 0) = 0.

⁽⁵⁾

If

𝑢

îs â^solution ôf

Δ𝑢 = 𝑓

^then ^so ^is

𝑢 + 𝑐

⁽

𝑐 ∈ ℝ

^), ^hene^the ^ODE ^for

𝑚 = 0

îs ^determined ûp ^to ân

additiveonstant.

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With

𝑚 ∕ = 0

^xed,^the^twoindependentsolutionsof

Δ𝑢 = 0

^are

𝑔 _± (𝜑, 𝑚) =

( sin(𝜑) cos(𝜑) + 1

) ±∣ 𝑚 ∣

= 𝑒 ^±∣ ^𝑚 ^∣ ^log ^cos(𝜑)+1 ^sin(𝜑) , 𝑚 ∈ ℤ ∖ { 0 } .

For

𝑚 = 0

^the^twoindependentsolutionsgive

ˆ

𝑢(𝜑, 0) = 𝐶 1 + 𝐶 2 log 1 − cos 𝜑 sin 𝜑 .

Ifweinsist,forinstane, that

𝑢 ∈ 𝐻 ¹ (Ω)

^,^weêliminate^the^logarithmi^termând^weôbtain^that

𝑢(𝜑, ˆ 0)

^is^a

onstantin

𝜑

^.

All theeigenvaluesof

Δ

âreôf^the^formôf

− 𝑛(𝑛 + 1)

^for

𝑛 = 0, 1, ...

^;ⁱⁿ ^partiular,^they^arenon-positive (and

Δ

^is^negativesemi-denite).

3.2 The Shwarz iteration for

Δ

with two latitudinal subdomains

ConsidertheShwarziteration(1). Weareinterestedin studyingtheerrorterms

𝑢 0 − 𝑢

^and

𝑣 0 − 𝑢

^where

Δ𝑢 = 𝑓

^. ^Observe^that

𝑢 𝑘 − 𝑢

^and

𝑣 𝑘 − 𝑢

âre^lassial^Shwarzîterates âs^perêquation^(1),^but^with

𝑓 = 0

^.

For this reason,it sues to analyzethe lassialShwarziteration with

𝑓 = 0

^, ^whih ^we^assume ^for^the

remainderofouranalysis.

UsingtheFouriertransformin

𝜃

^,^we^an^write

𝑢 ˆ 𝑘+2 (𝑏, 𝑚)

^expliitlyⁱⁿ ^terms^of

𝑢 ˆ 𝑘 (𝑏, 𝑚)

^. ^This^allows^us

toobtainaonvergenefatorestimate,whihwereallfrom [4℄.

Lemma3.1. The Shwarziteration onthesphere

Ω

partitionedalong twolatitudes

𝑏 < 𝑎

^onverges^(exept

for the onstantterm). The onvergene fator

∣ 𝑢 ˆ 𝑘+2 (𝑏, 𝑚)/ˆ 𝑢 𝑘 (𝑏, 𝑚) ∣

^is

𝐶(𝑚) =

( sin(𝑏) cos(𝑏) + 1

) 2 ∣ 𝑚 ∣ (

sin(𝑎) cos(𝑎) + 1

) − 2 ∣ 𝑚 ∣

< 1.

⁽⁶⁾

This onvergenefatordependsonthefrequeny

𝑚

^of

𝑢 𝑘

^on^the^latitude

𝑏

^.

AnanalysisthatislosertothenumerialalgorithmwouldbetoreplaetheontinuousFouriertransform

in

𝜃

^by^a^disrete^one.

Lemma3.2 (Semidisreteanalysis.). The Laplaian disretizedin

𝜃

^with

𝑛

^sample ^points:

Δ 𝑛 𝑢 = 𝑛 ² 4𝜋 ² sin ² 𝜑

( 𝑢

(

𝜑, 𝑗 + 1 2𝜋𝑛

)

− 2𝑢(𝜑, 𝑗) + 𝑢 (

𝜑, 𝑗 − 1 2𝜋𝑛

))

+ cot 𝜑𝑢 𝜑 + 𝑢 𝜑𝜑

⁽⁷⁾

leads toaShwarz iterationwhose onvergenefatoris

( sin(𝑏) cos(𝑏) + 1

) 2 ∣ 𝑚 ˜ ∣ (

sin(𝑎) cos(𝑎) + 1

) − 2 ∣ 𝑚 ˜ ∣

< 1

every twoiterations, where

˜

𝑚 ² = 𝑛 ²

4𝜋 ² (1 − cos(2𝜋𝑘/𝑛))

for the

𝑘

^th^frequeny.

Proof. WedoadisreteFouriertransformin

𝜃

^and^the^equation^to^solve^beomes

𝐿 𝑛 𝑢(𝜑, 𝑘) = ˆ 𝑛 ²

4𝜋 ² sin ² 𝜑 (cos(2𝜋𝑘/𝑛) − 1)ˆ 𝑢(𝜑, 𝑘) + cot 𝜑ˆ 𝑢 𝜑 (𝜑, 𝑘) + ˆ 𝑢 𝜑𝜑 (𝜑, 𝑘) = 0.

⁽⁸⁾

Bywriting

𝑚

âs ^notedⁱⁿ ^the^statementôf ^the^Lemma,^we^have^redued^this ^problem^to ^the ^previousône

andweanreuse Lemma3.1.

The two ontration onstants are verysimilar. For small valuesof

𝑚

^(ignoring

𝑚 = 0

^beause ^that

mode need notonvergeat all),the speedof onvergene is very poor. The overall onvergene fator as

measuredinthe

𝐿 ²

^normâlong^theînterfaesîs^given^by

sup

𝑚 ≥ 1

𝐶(𝑚) = 𝐶(1)

= 1 − 𝑂(𝑎 − 𝑏),

andsotheonvergenefatoroftheShwarziterationdeterioratesrapidlyas

𝑎 − 𝑏

^beomes^small.

(7)

4 Optimized Shwarz iteration for

Δ

with latitudinal boundaries Inthissetion,weproveTheorems2.1and2.2. Thisrequiresmanytehnialresultsandsoweproeedwith

asequeneofsimplelemmas.

Toanalyze the onvergene fator, as we did in Setion 3, weassume, without lossof generality, that

𝑓 = 0

^. ^Beause ^the^Laplaianôf âônstantîs ^zero, ^we ^will ^see^that ^theêrror^terms ônverge^to ônstant

funtions,whihisto say,theloalsolutionsonvergeupto anadditiveonstant.

Onekeytehniqueisto usethefat that,although

𝑢 0

^and

𝑣 0

^may^be^arbitraryⁱⁿ ^some^Sobolev^spaes,

𝑢 1

^and

𝑣 1

^are^harmoni ⁱⁿ

Ω 1

^and

Ω 2

^, respetively. This keyobservationallowsus to nditerations that onvergefaststarting from the seond step. Thisobservation hasanimpaton thenumerialsimulations.

Often, the rstiterations

𝑢 1

^and

𝑣 1

^are ^not^muh ^better ^than ^the ^initial ^guesses

𝑢 0

^and

𝑣 0

^, ^and^they ^are

sometimesevenworse.

Lemma 4.1 (Nonloal operator). If, for eah

𝑚

^,

𝜓(𝑚) = ˆ ∣ 𝑚 ∣ / sin 𝑎

^and

𝜉(𝑚) = ˆ −∣ 𝑚 ∣ / sin 𝑏

^,

Δ𝑢 1 = 0

ⁱⁿ

Ω 1

^and

Δ𝑣 1 = 0

ⁱⁿ

Ω 2

^, ^then

𝑢 2 = 0

^and

𝑣 2 = 0

^.

Proof. Let

𝑢 1 (𝜑, 𝜃) = ˆ 𝑢 1 (𝑎, 0)+ ∑

𝑘 ∕ =0 𝑢 ˆ 1 (𝑎, 𝑘)𝑔 + (𝜑, 𝑘)𝑒 ^𝑖𝑘𝜃

^and

𝑣 1 (𝜑, 𝜃) = ˆ 𝑣 1 (𝑎, 0)+ ∑

𝑘 ∕ =0 𝑣 ˆ 1 (𝑎, 𝑘)𝑔 ₋ (𝜑, 𝑘)𝑒 ^𝑖𝑘𝜃

^,

thentheboundaryvalueproblemforthenorthhemisphereyieldsforeah

𝑚 ∕ = 0

^:

ˆ

𝑢 2 (𝑎, 𝑚) (

𝜓(𝑚) + ˆ ∣ 𝑚 ∣ sin 𝑎

)

= ˆ 𝑣 1 (𝑎, 𝑚) (

𝜓(𝑚) ˆ − ∣ 𝑚 ∣ sin 𝑎

)

= 0,

⁽⁹⁾

and so

𝑢 ˆ 2 (𝑎, 𝑚) = 0

^. ^Likewise, ^we ^nd ^that

𝑣 ˆ 2 (𝑏, 𝑚) = 0

^for ^eah

𝑚 ∕ = 0

^. ^F^or

𝑚 = 0

^, ^we ^have ^that

ˆ

𝑢 2 (𝑎, 𝑚) = ˆ 𝑣 1 (𝑎, 𝑚)

^: ^the îteration ^does ^not âet ^the ônstant ^mode

𝑚 = 0

^. ^Sine ^the ^solution ^of ^the

Laplaeproblemisonlydeneduptoaonstant,theiterationhasonverged.

Theseondtangentialderivative

𝐷 ² _𝜃

^along^latitude

𝜑

^has^Fourier^transform

− 𝑚 ²

^. ^The^optimal^operators

are therefore the symmetri and positive denite square root of

− 𝐷 _𝜃 ²

^, ^multiplied ^by

1/ sin 𝑎

^or

1/ sin 𝑏

aordingto thelatitude. This normalizationonstantisrelatedtothelengthofthelatitudeat

𝜑 = 𝑎

^and

𝜑 = 𝑏

^,respetively,sinealatitude

𝜑

îsâîrle ôf^radius

𝑟 = sin 𝜑

^.

Next,westatetheonvergenefatoroftheoptimizedShwarziteration,if

𝜓

^and

𝜉

^are^not^these^nonloal

operators.

Lemma4.2. If

Δ𝑢 1 = 0

ⁱⁿ

Ω 1

^then

ˆ

𝑢 3 (𝑏, 𝑚) = 𝑢 ˆ 1 (𝑏, 𝑚) 𝜉(𝑚) sin ˆ 𝑏 + ∣ 𝑚 ∣ 𝜉(𝑚) sin ˆ 𝑏 − ∣ 𝑚 ∣

𝜓(𝑚) sin ˆ 𝑎 − ∣ 𝑚 ∣ 𝜓(𝑚) sin ˆ 𝑎 + ∣ 𝑚 ∣

𝑔 ₋ (𝑚, 𝑎) 𝑔 ₋ (𝑚, 𝑏)

𝑔 + (𝑚, 𝑏) 𝑔 + (𝑚, 𝑎) .

Theproof isasimpleomputation usingequation(9). Wenowlookforgoodonvolutionkernels

𝜉

^and

𝜓

^so ^that^the^the^onvergene^fator

𝜅(𝜓, 𝜉, 𝑚, 𝑎, 𝑏) =

𝜅 ₁

z }| {

𝜉(𝑚) sin𝑏 ˆ + ∣𝑚∣

𝜉(𝑚) sin𝑏 ˆ − ∣𝑚∣

𝜅 ₂

z }| {

𝜓(𝑚) sin ˆ 𝑎 − ∣𝑚∣

𝜓(𝑚) sin ˆ 𝑎 + ∣𝑚∣

𝜅 3

z }| { 𝑔 − (𝑚, 𝑎) 𝑔 − (𝑚, 𝑏)

𝜅 4

z }| { 𝑔 + (𝑚, 𝑏)

𝑔 + (𝑚, 𝑎) ,

⁽¹⁰⁾

with

𝑚 ∈ ℤ ∖ { 0 }

^,îsâs^smallâs^possible. ^Forâ^given^hoieôf

𝜓 ˆ

^and

𝜉 ˆ

^,^the^onvergene^fator^as^measured

inthe

𝐿 ²

^norm^along^the^interfae

𝜑 = 𝑏

^will ^be^given^by

sup _𝑚 _∈ℤ∖{ ₀ _} ∣ 𝜅(𝜓, 𝜉, 𝑚, 𝑎, 𝑏) ∣ .

Theonvergenefatorestimate(10)alsoleadstothefollowingresult:

Corollary 4.3. The iteration (2) is onvergent (modulo the onstant mode)if

𝜓

^and

− 𝜉

^are ^onvolution

operatorsthatarepositive denite,regardless ofoverlap.

Proof. Let

𝜓(𝑚) ˆ > 0

^and

𝜉(𝑚) ˆ < 0

^,^and^let

𝜅 1 , 𝜅 2 , 𝜅 3 , 𝜅 4

^be^asⁱⁿ ^(10). ^Sine

sin 𝑎 > 0

^and

sin 𝑏 > 0

^,^we^see

that

∣ 𝜅 1 ∣ < 1

^and

∣ 𝜅 2 ∣ < 1

^. Furthermore,

𝜅 3 𝜅 4 =

( sin(𝑏) cos(𝑏) + 1

) 2 ∣ 𝑚 ∣ (

sin(𝑎) cos(𝑎) + 1

) − 2 ∣ 𝑚 ∣

=

⎛

⎜ ⎜

⎜ ⎝

≤ 1

z }| { ( sin(𝑏)

sin(𝑎) )

≤ 1

z }| { ( cos(𝑎) + 1

cos(𝑏) + 1 )

⎞

⎟ ⎟

⎟ ⎠

2 ∣ 𝑚 ∣

≤ 1,

wherewehaveusedthat

𝑏 < 𝑎

^. ^Hene,

∣ 𝜅(𝜓, 𝜉, 𝑚, 𝑎, 𝑏) ∣ < 1

^for^every

𝑚 ∕ = 0

^.

(8)

By omparison, in [27℄, it was shown that if

𝜉

^and

𝜓

^are ^Robin transmission onditionsfor a domain deomposition with relativelyuniform overlap,then the iterationonvergesso long as

𝜉

^and

𝜓

^are ^both

positivedenite.

Our analysis willproeed by onsideringthe ontrationonstant (10)forthevarious hoiesof trans-

missiononditions

𝜉

^and

𝜓

^. ^Weônsider^rst^theûseôf^Robintransmissiononditionsand proveTheorem 2.1. Seond,weproveTheorem2.2forseondordertangentialtransmissiononditions.

4.1 Robin or OO0 transmission onditions

Let

𝜓𝑢 = 𝑐𝑢

^and

𝜉𝑢 = 𝑑𝑢

^,^then

𝜅 𝑂𝑂0 (𝜓, 𝜉, 𝑚, 𝑎, 𝑏) = 𝑑 sin 𝑏 + ∣ 𝑚 ∣ 𝑑 sin 𝑏 − ∣ 𝑚 ∣

𝑐 sin 𝑎 − ∣ 𝑚 ∣ 𝑐 sin 𝑎 + ∣ 𝑚 ∣

𝑔 ₋ (𝑚, 𝑎) 𝑔 ₋ (𝑚, 𝑏)

𝑔 + (𝑚, 𝑏) 𝑔 + (𝑚, 𝑎)

= 𝜅(𝑐, 𝑑, 𝑚, 𝑎, 𝑏).

For

𝑎, 𝑏

^xed,^we^now^need^to^ompute^the^hoies^of

𝑐, 𝑑

^minimizing

𝜑(𝑐, 𝑑) = sup

𝑚 𝜅 𝑂𝑂0 (𝑐, 𝑑, 𝑚, 𝑎, 𝑏) ² ,

where wehaveintrodued asquare to simplify somealulations; theatual ontration onstantwill be

givenby

√ 𝜑(𝑐, 𝑑)

^. ^Tô^simplifyôur^notation,îtîsônvenient^to^doâ^hangeôf^variables. ^We^set

𝑥 = − 𝑑 sin 𝑏

^,

𝑦 = 𝑐 sin 𝑎

^and

𝐿 = − 2 log

( 1 + cos 𝑎 sin 𝑎

sin 𝑏 1 + cos 𝑏

)

≥ 0,

andrewritingthe

𝑔 _±

^terms^asexponentials,theoptimizationproblembeomes Minimize

𝜑(𝑥, 𝑦)

^where

𝜑(𝑥, 𝑦) = sup

𝑚 ∈{ 1,2,... }

𝐹 (𝑥, 𝑦, 𝑚) 𝐹 (𝑥, 𝑦, 𝑚) =

𝑥 − 𝑚 𝑥 + 𝑚

𝑦 − 𝑚 𝑦 + 𝑚 𝑒 ⁻ ^𝐿𝑚

2

(11)

= 𝐹 𝐿 (𝑥, 𝑦, 𝑚).

The onvergene fator every other step for partiular parameter hoies

𝑥

^and

𝑦

^is

√

𝜑(𝑥, 𝑦)

^. ^The ^ase

𝐿 = 0

^only^ours ^when

𝑎 = 𝑏

^,ⁱⁿ ^whih ^ase

𝜑(𝑥, 𝑦) = 1

^for^all

𝑥, 𝑦 ≥ 1

^and ^then^we^will ^need^to^take^into

aountertaindisretizationparameterstoobtainaonvergenefatorestimate. Thisasewillbetreated

separately.

WenowusetwodierentanalysestoomputeoptimizedRobinparameters

𝑐

^and

𝑑

^. ^First,^we^make^the

simplifyingassumption that

𝑥 = 𝑦

^,âllowingûs^toôptimizeînstead^theôbjetive^funtion

max 𝑚 ∣ (𝑥 − 𝑚)/(𝑥 + 𝑚)𝑒 ⁻ ^𝐿 ² ^𝑚 ∣ ² .

This situation is interesting beause this an lead to an algorithm where all subdomains do the same

thing. Thisisthemethodwhihmostobviouslygeneralizesto manysubdomains. Seond,weonsiderthe

expression

𝐹 (𝑥, 𝑦, 𝑚)

^with^both^parameterssimultaneously. Thisisalsointeresting,beauseinthissituation, the iteration does something dierent on eah subdomain. We will see that, although this approah is

diulttogeneralizetomanysubdomains,itleadsto substantiallyimprovedonvergenefators.

4.2 OO0 one-sided analysis,

𝐿 > 0

.

Weanwrite

𝐹 𝐿 (𝑥, 𝑦, 𝑚) = 𝐹 𝐿 (𝑥, 𝑚)𝐹 𝐿 (𝑦, 𝑚)

^(f. ^(11)),^where^we^dene

𝐹 𝐿 (𝑥, 𝑚) = 𝐹(𝑥, 𝑚) =

𝑥 − 𝑚 𝑥 + 𝑚 𝑒 ⁻ ^𝐿 ² ^𝑚

2 .

(9)

Werstlookattheproblemofomputingtheoptimalparameter

𝑥 0

^minimizing

𝜑 𝐿 (𝑥) = sup

𝑚 ∈{ 1,2,... }

𝐹 𝐿 (𝑥, 𝑚),

sine it is simpler. The solution

𝑥 0

^minimizing

𝜑 𝐿 (𝑥)

^leads ^to ^the ^hoies

𝑥 = 𝑦 = 𝑥 0

^, ^whih ^minimizes

𝜑 𝐿 (𝑥, 𝑦 )

^subjet^to^the^symmetry^onstraint

𝑥 = 𝑦

^. ^The^onvergene^fator^for^a^partiular^parameter^hoie

𝑥

^is

√

𝜑(𝑥, 𝑥) = 𝜑 𝐿 (𝑥)

^.

We now deal with the various spetra over whih we an optimize. Our ontinuous problem has a

frequenyvariable

𝑚 ∈ ℤ

^,^but^sine^the

𝑚 = 0

^frequeny^need^not^onverge,^and^sine

𝐹 𝐿 (𝑥, − 𝑚) = 𝐹 𝐿 (𝑥, 𝑚)

^,

wehavealreadysimpliedourproblembyonsideringthefrequenies

𝑚 ∈ { 1, 2, . . . }

^. ^Beauseôfôurêventual

disretization,wewillwanttoonsiderthefrequenydomain

{ 1, 2, ..., 𝑁 }

^,^where

𝑁

^is^the^highest^frequeny

thatan beresolvedontheinterfae,givenourgrid. However,beauseofthediulties inoptimizingover

disretesets, wealso onsider thepossibilityof allowing

𝑚

^to^vary ⁱⁿ ^the^intervals

[1, ∞ )

^as^well^as

[1, 𝑁]

^.

For tehnialreasons,itisalsoonvenienttoinlude

𝑚 = ∞

ⁱⁿ^ouralulations.

Let

2 < 𝑁 < ∞

^and^dene

𝜑 ^(𝑐) _𝐿 (𝑥) = sup

𝑚 ∈ [1, ∞ ]

𝐹 𝐿 (𝑥, 𝑚), (𝐼 ^(𝑐) = [1, ∞ ]), 𝜑 ^(𝑑) _𝐿 (𝑥) = sup

𝑚 ∈{ 1,2,... }

𝐹 𝐿 (𝑥, 𝑚), (𝐼 ^(𝑑) = { 1, 2, ..., ∞} ), 𝜑 ^(𝑐𝑏) _𝐿 (𝑥) = sup

𝑚 ∈ [1,𝑁]

𝐹 𝐿 (𝑥, 𝑚), (𝐼 ^(𝑐𝑏) = [1, 𝑁]), 𝜑 ^(𝑑𝑏) _𝐿 (𝑥) = sup

𝑚 ∈{ 1,2,...,𝑁 }

𝐹 𝐿 (𝑥, 𝑚), (𝐼 ^(𝑑𝑏) = { 1, 2, ..., 𝑁 } ).

Aordingtoourdenitions, forall

𝑥

^,

𝜑 ^(𝑑) _𝐿 (𝑥) = 𝜑 𝐿 (𝑥)

^. ^Theôther ^funtionsâre^similar^but ^maximizeôver

adierentset offrequenies.

One of the main toolsfor solvingmin-max problems is theenvelopetheorem, whih westate herefor

onveniene.

Theorem 4.4 (The EnvelopeTheorem). Let

𝑈

^be ^open ⁱⁿ

ℝ ^𝑛

^and

𝜑 : 𝑈 → 𝑅

^,

𝛾 ∈ ℝ ^𝑛

^,

∣∣ 𝛾 ∣∣ = 1

^. ^For

𝑥 ∈ 𝑈

^, ^dene^the ^one-sided ^diretional ^derivative

𝐷 𝛾 𝜑(𝑥)

^by

𝐷 𝛾 𝜑(𝑥) = lim

𝜖 ↓ 0 ⁺

𝜑(𝑥 + 𝜖𝛾) − 𝜑(𝑥)

𝜖 ,

if the(one-sided)limit exists. Let

𝑉

^be^a^ompat ^metri^spae. ^Let

𝐹 : 𝑈 × 𝑉 → ℝ

(𝑥, 𝑦) 7→ 𝐹 (𝑥, 𝑦)

beontinuousandassumethatthepartialgradient

𝐹 𝑥 (𝑥, 𝑦)

^exists^for^all

𝑥, 𝑦 ∈ 𝑈 × 𝑉

^and^variesontinuously in

(𝑥, 𝑦)

^. ^Dene

𝜑(𝑥) = max

𝑦 ∈ 𝑉 𝐹(𝑥, 𝑦),

𝑌 (𝑥) = { 𝑦 ∈ 𝑉 ∣ 𝐹 (𝑥, 𝑦) = 𝜑(𝑥) } ∕ = ∅ .

Then,

1. Forall

𝑥 ∈ 𝑈

^and

𝛾 ∈ ℝ ^𝑛 , ∣∣ 𝛾 ∣∣ = 1

^,

𝐷 𝛾 𝑢(𝑥)

êxistsândîs^given ^by^the ^formula

𝐷 𝛾 𝜑(𝑥) = max

𝑦 ∈ 𝑌 (𝑥)

∑ 𝑛

𝑖=1

𝛾 𝑖 𝐹 𝑥 𝑖 (𝑥, 𝑦).

⁽¹²⁾

2. Let

𝑥 ˜ ∈ 𝑈

^. ^If

𝑌 (˜ 𝑥) = { 𝑦(˜ 𝑥) }

^is^a^single^point^then

𝐷 𝛾 𝜑(˜ 𝑥) =

∑ 𝑛

𝑖=1

𝛾 𝑖 𝐹 𝑥 𝑖 (˜ 𝑥, 𝑦)

⁽¹³⁾

(10)

and

𝜑

^is^(fully) dierentiableat

𝑥 ˜

^. ^If furthermore

𝑌 (𝑥) = { 𝑦(𝑥) }

îsâ^singleton ^for âll

𝑥 ∈ 𝑂 ⊂ 𝑈

^an

open set,then

𝜑

^isontinuouslydierentiablein

𝑂

^.

This immediatelyleadstoanequiosillationproperty.

Lemma4.5. For

𝑗 ∈ { 𝑐, 𝑑, 𝑐𝑏, 𝑑𝑏 }

^, ^there^is^a^minimizer

𝑥 𝑗

^of

𝜑 ^(𝑗) (𝑥)

^,^and

∣ 𝑀 ^(𝑗) (𝑥 𝑗 ) ∣ ≥ 2

^, ^wher^e

𝑀 ^(𝑗) (𝑥) = { 𝑚 ∈ 𝐼 ^(𝑗) ∣ 𝜑 ^(𝑗) (𝑥) = 𝐹 𝐿 (𝑥, 𝑚) }

^.

Proof. Wemustplaeourselvesinthehypothesesoftheenvelopetheorem. Eahset

[1, ∞ ], { 1, 2, ..., ∞} , [1, 𝑁 ], { 1, 2, ..., 𝑁 }

isaompatmetrispae(for

𝐼 ^(𝑐)

^and

𝐼 ^(𝑑)

^one^may^use^the^metri

𝑑(𝑥, 𝑦) = ∣ 𝑥 ⁻ ¹ − 𝑦 ⁻ ¹ ∣

^). ^Sine^the^funtion

𝜑 ^(𝑗)

îs â ôntinuous ^funtion ôn ^the ômpat ^metri ^spae

𝐼 ^(𝑗)

^, ît ^must ^have â ^minimum. ^The ôptimal

parameter

𝑥 𝑗

^isⁱⁿ

[0, ∞ )

^so^we^take

𝑈 = ( − ¹ 2 , ∞ )

^,ând^the^hypothesesôf^theênvelope^theoremâre^satised.

Assumethat

𝑀 ^(𝑗) (𝑥 𝑗 ) = { 𝑚 ˜ }

îsâ^singleton^(thereîs^noequiosillation). Then,bytheenvelopetheorem, thetwo-sidedderivative

𝐷 𝑥 𝜑 ^(𝑗) (𝑥 𝑗 )

^exists^and^sine

𝜑 ^(𝑗) (𝑥 𝑗 )

^is^minimal,^the^derivative^must^be^zero. ^Using

formula(13),weobtain

0 = 𝐹 𝑥 (𝑥 𝑗 , 𝑚) = ˜ − 4 ˜ 𝑚 𝑚 ˜ − 𝑥 𝑗

( ˜ 𝑚 + 𝑥 𝑗 ) ³ 𝑒 ⁻ ^2𝐿 ^𝑚 ^˜ .

Sine

𝑥 𝑗 , 𝑚 > ˜ 0

^, ^it^must ^be^that

𝑚 ˜ = 𝑥 𝑗

^. ^Hene,

𝜑 ^(𝑗) (𝑥 𝑗 ) = 𝐹 (𝑥 𝑗 , 𝑚) = 0 ˜

^. ^But^we ^know^that

𝜑 ^(𝑗) (𝑥 𝑗 ) ≥ 𝐹 (𝑥, 𝑥 + 1) > 0

^.

Nowweharaterizetheuniqueminimizerforthease

𝑗 = 𝑐

^.

Lemma4.6. Thereisaunique

𝑥 𝑐

^minimizing

𝜑 ^(𝑐) _𝐿 (𝑥)

ândîtîs^the ûnique^root

𝜓(𝑥 𝑐 ) = 0

^of

𝜓(𝑥) = 𝜓(𝑥, 𝐿) =

( 𝑚(𝑥) − 𝑥 𝑚(𝑥) + 𝑥

) 2

𝑒 ⁻ ^{𝐿𝑚(𝑥)} − ( 1 − 𝑥

1 + 𝑥 ) 2

𝑒 ⁻ ^𝐿 ,

where

𝑚(𝑥) =

√ 4𝑥 𝐿 + 𝑥 ²

^.

Proof. Beause any minimizer

𝑥 𝑐

^is ⁱⁿ

[1, ∞ )

^, ^the ^only^loal ^maxima ^of

𝐹 𝐿 (𝑥, 𝑚)

^as

𝑚

^runs ⁱⁿ

[1, ∞ ]

^are

at

𝑚 = 1

^and

𝑚 = 𝑚(𝑥) = √

4𝑥

𝐿 + 𝑥 ²

^and ^so

𝑀 ^(𝑐) (𝑥 𝑐 ) ⊆ { 1, 𝑚(𝑥 𝑐 ) }

^. ^By ^Lemma ^4.5, ^we ^know ^that

#𝑀 ^(𝑐) (𝑥 𝑐 ) ≥ 2

^so

𝑀 ^(𝑐) (𝑥 𝑐 ) = {

1,

√ 4𝑥 𝑐

𝐿 + 𝑥 ² _𝑐 }

.

Hene,

𝜓(𝑥 𝑐 ) = 0

^. ^This^shows^theequiosillation

𝜓(𝑥) = 0

^.

Wenowshowtheuniquenessofthesolutionoftheequiosillationproblem. Tothatend, let

𝑔(𝑥) = 𝐹 (𝑥, 𝑚(𝑥)),

𝑔 ^′ (𝑥) = 𝐹 𝑥 (𝑥, 𝑚(𝑥)) + 𝐹 𝑚 (𝑥, 𝑚(𝑥))𝑚 ^′ (𝑥)

= 𝐹 𝑥 (𝑥, 𝑚(𝑥))

= − 4𝑚(𝑥) 𝑚(𝑥) − 𝑥

(𝑚(𝑥) + 𝑥) ³ 𝑒 ⁻ ^{𝐿𝑚(𝑥)} < 0,

and

ℎ(𝑥) = 𝐹 (𝑥, 1), ℎ ^′ (𝑥) = − 4 1 − 𝑥

(1 + 𝑥) ³ 𝑒 ⁻ ^𝐿 > 0.

Hene,

𝜓 ^′ = 𝑔 ^′ − ℎ ^′ < 0

^and

𝜓

^may^not^have^multiple^zeros.

Thisresultgivessomemoreinformationforthease

𝑗 = 𝑑

^(that^is,

𝐼 ^(𝑑) = { 1, 2, ..., ∞}

^).^We^must^assume

that

𝐿

îs^not^too^big,^butôur^fousîs^theâse^where^thereîs^littleôverlap^so^thisîs^notâ^problem. ^The^speial

point

𝑚(𝑥)

ôntinues^to^playânîmportant^role. ^Notie^that

𝑚 ^′ (𝑥) > 1

^and^that

𝑚(1) = √

4 + 𝐿/ √

𝐿 > 1

^so

that, forall

𝑥 > 1

^,

𝑚(𝑥) > 𝑚(1) + (𝑥 − 1) > 𝑥

^.

(11)

Lemma 4.7. Let

𝐿 < 4/3

^and ^let

𝑥 𝑐

^minimize

𝜑 ^(𝑐) _𝐿 (𝑥)

^and

𝑥 𝑑

^minimize

𝜑 ^(𝑑) _𝐿 (𝑥)

^. ^Then

1 ∈ 𝑀 ^(𝑑) (𝑥 𝑑 ) ⊆ { 1, ⌈ 𝑚(𝑥 𝑑 ) ⌉ , ⌊ 𝑚(𝑥 𝑑 ) ⌋}

^and

𝑥 𝑐 − 1 ≤ 𝑥 𝑑 ≤ 𝑥 𝑐

^.

Proof. Beause

𝑚 = 𝑚(𝑥 𝑑 )

îs^theônly^loal^maximumôf

𝐹 𝐿 (𝑥, 𝑚)

^,^we^know^that

𝑀 ^(𝑑) (𝑥 𝑑 ) ⊆ { 1, ⌈ 𝑚(𝑥 𝑑 ) ⌉ , ⌊ 𝑚(𝑥 𝑑 ) ⌋}

^,

and we also knowfrom Lemma 4.5 that

#𝑀 ^(𝑑) (ˆ 𝑥) ≥ 2

^. ^Assume ^that

𝑀 ^(𝑑) (ˆ 𝑥) = {⌈ 𝑚(ˆ 𝑥) ⌉ , ⌊ 𝑚(ˆ 𝑥) ⌋}

^. ^The

hypothesis on

𝐿

^ombined^with^the^fat ^that

𝑥 𝑑 ≥ 1

^guarantees^that

𝑚(𝑥 𝑑 ) > 𝑥 𝑑 + 1

^so

𝑥 𝑑 < ⌊ 𝑚(𝑥 𝑑 ) ⌋

^. ^We

annowusetheone-sidedderivativeoftheEnvelopeTheorem,obtainingthat

𝐷 (1) 𝜑 ^(𝑑) (𝑥 𝑑 ) = max

𝑚 ∈{⌈ 𝑚(𝑥 𝑑 ) ⌉ , ⌊ 𝑚(𝑥 𝑑 ) ⌋} 𝐹 𝑥 (𝑥 𝑑 , 𝑚)

= max

𝑚 ∈{⌈ 𝑚(𝑥 _𝑑 ) ⌉ , ⌊ 𝑚(𝑥 _𝑑 ) ⌋} − 𝑚 𝑚 − 𝑥 𝑑

(𝑚 + 𝑥 𝑑 ) ³ 𝑒 ⁻ ^𝐿𝑚 < 0.

Therefore,

1 ∈ 𝑀 ^(𝑑) (ˆ 𝑥)

^.

Clearly, wemusthave

𝜑 ^(𝑑) (𝑥 𝑑 ) ≤ 𝜑 ^(𝑐) (𝑥 𝑐 )

^. ^If

𝑥 𝑑 > 𝑥 𝑐

^then

𝜑 ^(𝑑) (𝑥 𝑑 ) ≥ 𝐹(𝑥 𝑑 , 1) = 𝜑 ^(𝑐) (𝑥 𝑑 ) > 𝜑 ^(𝑐) (𝑥 𝑐 )

^,

henewemust have that

𝑥 𝑑 ≤ 𝑥 𝑐

^. ^If

𝑥 𝑑 < 𝑥 𝑐 − 1

^, ^then ⁱⁿ ^view^of ^the ^estimate

𝑚 ^′ (𝑥) > 1

^, ^we^have ^that

𝑚(𝑥 𝑑 ) < 𝑚(𝑥 𝑐 ) − 1

^. ^Let

𝑚 0 = ⌊ 𝑚(𝑥 𝑐 ) ⌋

^and^nd^the ^unique

𝑥 0 ∈ (1, 𝑚 0 )

^suh^that

𝑚(𝑥 0 ) = 𝑚 0

^. ^We^have

that

𝑥 𝑑 ∈ (1, 𝑥 0 ) ⊂ (1, 𝑚 0 )

^. ^F^or

𝑥 ∈ (1, 𝑚 0 )

^,^we^have^that

𝐹 𝑥 (𝑥, 𝑚 0 ) < 0

^. ^Hene,

𝜑 ^(𝑑) (𝑥 𝑑 ) ≥ 𝐹 (𝑥 𝑑 , 𝑚 0 ) >

𝐹 (𝑥 0 , 𝑚 0 ) = 𝜑 ^(𝑐) (𝑥 0 ) > 𝜑 ^(𝑐) (𝑥 𝑐 )

^.

As aresult of Lemma 4.7, we see that using thedisrete frequenyspetrum

{ 1, 2, ..., ∞}

^or ^using ^the

ontinuousfrequenyspetrum

[1, ∞ ]

^resultsⁱⁿapproximatelythesameRobinparameter.

We would liketo givea formula for the optimized Robin parameter, but the equiosillation property

annotbesolvedexpliitly. However,wean solveitapproximatelyfor small

𝐿

^. ^This^further ^allows^us ^to

onsiderthespetrum

[1, 𝑁 ]

^. ^If

𝑥 𝑐𝑏

^minimizes

𝜑 ^(𝑐𝑏) (𝑥)

^then

𝑀 ^(𝑐𝑏) (𝑥 𝑐𝑏 ) ⊆ { 1, 𝑁, 𝑚(𝑥 𝑐𝑏 ) }

^. ^When

𝐿

^is^not^too

smallomparedto

𝑁 ⁻ ¹

^,^we^now^show^that

𝑀 ^(𝑐𝑏) (𝑥 𝑐𝑏 ) = { 1, 𝑚(𝑥 𝑐𝑏 ) }

^. ^This ^implies^that, ^for^those^values^of

𝐿

^and

𝑁

^,^the^optimized^parameter^for^the^frequeny^spetrum

[1, 𝑁]

îs^the^sameâs^theône^for^the^frequeny

spetrum

[1, ∞ ]

^.

Lemma4.8. The optimizedhoie

𝑥 𝑐

^for

𝜑 ^(𝑐) (𝑥 𝑐 )

^is^asymptoti^to

𝐿 ⁻ ¹ ³

^:

𝐿 lim → 0 ⁺ 𝑥 𝑐 𝐿 ¹ ³ = 1.

If wehoose

𝑥 ˆ 𝑐 = 𝐿 ⁻ ¹ ³

âsôurôptimized^parameter, ^the ônvergene ^fatorêvery ôther^stepîs

𝐹 (ˆ 𝑥 𝑐 , 1) = 1 − 4𝐿 ¹ ³ + 𝑂(𝐿 ² ³ ).

⁽¹⁴⁾

Let

𝛼 > − 3/2

^. ^If

𝐿 = 𝐿(𝑁) > 𝑁 ^𝛼

^then ^for ^all

𝑁

^suiently ^large,

𝑚(𝑥 𝑐 ) < 𝑁

^, ^and ^for ^suh^values ^of

𝑁

and

𝐿

^, ^we^have ^that

𝑥 𝑐 = 𝑥 𝑐𝑏

^.

Proof. Wetrytosolve

𝐹 (𝑥, 𝑚(𝑥)) − 𝐹 (𝑥, 1) = 0

^by^writing

𝑥

âsâ^powerôf

𝐿

^. ^Let

𝑥 = 𝐶𝐿 ^𝛽

^,^then^we^get

0 = 𝜓 =

( √

𝐶𝐿 ^𝛽 ⁻ ¹ + 𝐶 ² 𝐿 ^2𝛽 − 𝐶𝐿 ^𝛽

√ 𝐶𝐿 ^𝛽 ⁻ ¹ + 𝐶 ² 𝐿 ^2𝛽 + 𝐶𝐿 ^𝛽 ) 2

𝑒 ⁻ ^𝐿 √

4𝐶𝐿 ^𝛽− ¹ +𝐶 ² 𝐿 ^2𝛽

−

( 1 − 𝐶𝐿 ^𝛽 1 + 𝐶𝐿 ^𝛽

) ²

𝑒 ⁻ ^𝐿 ,

⁽¹⁵⁾

whih is dened for

𝐿 > 0

^and

𝐶 > 0

^. ^We ^want ^to ^take ^a ^series ^expansion ^for

𝜓

^, ^and ^we ^nd ^that ^for

𝛽 = − 1/3

^,

𝜓 = ( 4

𝐶 − 4 √ 𝐶

)

𝐿 ³ ¹ + 𝑂(𝐿 ² ³ ).

Wesee that

𝐺(𝑥) = 𝐹 (𝑥, 𝑚(𝑥)) − 𝐹(𝑥, 1) < 0

^for ^suiently^small

𝐿

^if^we^hoose

𝑥 = 𝐶𝐿 ⁻ ¹ ³

^with

𝐶 > 1

^,

and

𝐺(𝑥) > 0

^for^all^suiently^small

𝐿

^if

𝐶 < 1

^. ^Hene,

𝑥 𝑐 𝐿 ¹ ³ → 1

^as

𝐿 → 0 ⁺

^.

Toshowthat

𝑥 𝑐 = 𝑥 𝑐𝑏

^,^it^sues^to^show^that

𝑚(𝑥 𝑐 ) < 𝑁,

⁽¹⁶⁾

forsuientlysmall

𝐿

^. ^We^substitute

𝑥 = 𝐿 ⁻ ¹ ³

^and^then

𝐿 = 𝑁 ^𝛼

^into

𝑚(𝑥)

^obtaining

𝑚(𝑥) =

√

4𝑁 ⁻ ⁴ ³ ^𝛼 + 𝑁 ⁻ ² ³ ^𝛼 .

Hene if

𝛼 > − 3/2

^,

𝑚(𝑥)/𝑁 → 0

^as

𝑁 → ∞

^and ^so

𝑚(𝑥) < 𝑁

^for ^large

𝑁

^. Conversely, if

𝛼 < − 3/2

^,

𝑚(𝑥)/𝑁 → ∞

^and

𝑚(𝑥) > 𝑁

^for^large

𝑁

^.

(12)

Thease

{ 1, 2, ..., 𝑁 }

^followsimmediately.

Corollary 4.9. Let

𝛼 > − 2 ³

^. ^Let

𝐿 = 𝐿(𝑁 ) > 𝑂(𝑁 ^𝛼 )

^,

𝐿 < ⁴ ₃

^and ^let

𝑥 𝑐𝑏

^minimize

𝜑 ^(𝑐𝑏) (𝑥)

^. ^If

𝑥 𝑑𝑏

minimizes

𝜑 ^(𝑑𝑏) (𝑥)

^then^for ^all

𝑁

^suiently ^large,

1 ∈ 𝑀 (𝑥 𝑑𝑏 ) ⊂ { 1, ⌊ 𝑚(𝑥 𝑐 ) ⌋ , ⌈ 𝑚(𝑥 𝑐 ) ⌉}

^.

Proof. Say that

𝑥 𝑑

^minimizes

𝜑 ^(𝑑) (𝑥)

^. ^By ^lemma ^4.7, ^we ^have ^that

𝑀 ^(𝑑) (𝑥 𝑑 ) ⊆ { 1, ⌈ 𝑚(𝑥 𝑐 ) ⌉ , ⌊ 𝑚(𝑥 𝑐 ) ⌋}

^,

where

𝑥 𝑐

^minimizes

𝜑 ^(𝑐) (𝑥)

^. ^Lemma ^4.8 ^gives

⌈ 𝑚(𝑥 𝑐 ) ⌉ < 𝑁

^for ^large

𝑁

^, ^hene

𝑀 ^(𝑑) (𝑥 𝑑 ) ⊂ { 1, ..., 𝑁 }

^. ^If

𝑥 𝑑𝑏 > 𝑥 𝑑

^then

𝐹 (𝑥 𝑑𝑏 , 1) > 𝐹 (𝑥 𝑑 , 1)

^and

𝑥 𝑑𝑏

îs ^not â ^minimizer. Îf

𝑥 𝑑𝑏 < 𝑥 𝑑

^and ^if

1 ∕ = 𝑚 𝑑 ∈ 𝑀 ^(𝑑) (𝑥 𝑑 )

then

𝜑 ^(𝑑𝑏) (𝑥 𝑑𝑏 ) ≥ 𝐹(𝑥 𝑑𝑏 , 𝑚 𝑑 ) ≥ 𝐹 (𝑥 𝑑 , 𝑚 𝑑 ) = 𝜑 ^(𝑑) (𝑥 𝑑 )

^and

𝑥 𝑑𝑏

^is ^not^a ^minimum. ^Hene

𝑥 𝑑 = 𝑥 𝑑𝑏

^for ^all

suientlylarge

𝑁

^.

Notethat theasymptotionvergenefator(14)isapproximatelyindependentofthespetrumweuse,

andsowemayusetheoptimizedparameter

𝑥 ˆ 𝑐 = 𝐿 ⁻ ¹ ³

ⁱⁿ ^allsituations.

This ompletestheproofoftherstpartofTheorem2.1.

4.3 Two-parameter OO0 analysis,

𝐿 > 0

.

Returningtoformula(11),wenowomputetheoptimalparameters

𝑥

^and

𝑦

^without^assuming^that

𝑥 = 𝑦

^.

Calulationsinthissettingaremoreompliatedthanintheone-parameterase,so wetreatonlytheases

of thespetrum

[1, ∞ ]

^and

[1, 𝑁]

^. ^We ^an ^get ^arudimentary equiosillationpropertyout of theenvelope theorem.

Lemma4.10. Let

𝐿 > 0

^and

𝐹 (𝑥, 𝑦, 𝑚) =

( 𝑥 − 𝑚 𝑥 + 𝑚

𝑦 − 𝑚 𝑦 + 𝑚 𝑒 ⁻ ^𝐿𝑚

) 2

, 𝜑(𝑥, 𝑦) = sup

𝑚 ∈ [1, ∞ ]

𝐹 (𝑥, 𝑦, 𝑚),

𝑀 (𝑥, 𝑦) = { 𝑚 ∈ [1, ∞ ] ∣ 𝐹 (𝑥, 𝑦, 𝑚) = 𝜑 ^(𝑗) (𝑥, 𝑦) } .

Then

𝜑(𝑥, 𝑦)

^has^a ^minimum

(𝑥 0 , 𝑦 0 )

^. ^In^addition,

#𝑀 ^(𝑗) (𝑥 0 , 𝑦 0 ) ≥ 2

^and

min(𝑥 0 , 𝑦 0 ) ≥ 1

^.

Proof. Note that

𝜑(1, 1) < 1

^. ^Sine

𝜑(𝑥, 𝑦) ≥ 𝐹 (𝑥, 𝑦, 1)

^and ^sine

𝐹 (𝑥, 𝑦, 1)

^tends ^to

1

^as

(𝑥, 𝑦)

^tends ^to

∞

^, ^there^is ^an

𝑀 < ∞

^suh ^that, ^if

(𝑥, 𝑦) ∈ / [1, 𝑀 ] × [1, 𝑀 ]

^, ^then

𝜑(𝑥, 𝑦) > 𝜑(1, 1)

^. ^Hene,

𝜑

^must ^have^a

minimumin

[1, 𝑀 ] × [1, 𝑀 ]

^.

We now plae ourselves under the hypotheses of the Envelope Theorem. We have that

(𝑥 0 , 𝑦 0 ) ∈ ( ¹ ₂ , ∞ ) ² = 𝑈

^and^the^hoie

𝑉 = [1, ∞ ]

^does^the^trik.

If

𝑀 ^(𝑗) (𝑥 0 , 𝑦 0 ) = { 𝑚 }

^is^a^singleton,^then

𝜑(𝑥, 𝑦)

^isdierentiableat

𝑥 0 , 𝑦 0

^and

0 = ∇ 𝜑(𝑥 0 , 𝑦 0 )

= ( ∂𝐹

∂𝑥 (𝑥 0 , 𝑦 0 , 𝑚), ∂𝐹

∂𝑦 (𝑥, 𝑦, 𝑚) )

.

Itsuestolook atthepartialin

𝑥

^,

0 = − 4𝑚 𝑚 − 𝑥 (𝑚 + 𝑥) ³

( 𝑚 − 𝑦 𝑚 + 𝑦

) 2

𝑒 ⁻ ^2𝐿𝑚 ,

toonludethateither

𝑚 = 𝑥

^or

𝑚 = 𝑦

^. Înêitherâse,

𝜑(𝑥, 𝑦) = 𝐹 (𝑥, 𝑦, 𝑚) = 0

^,^whih ^is^absurd.

In theone-sided ase,this result was suient beause theonly remainingpossibilitywas tohavetwo

pointsequiosillating. However,in thetwo-sidedase,weanequiosillatetwoorthree points,but lemma

4.10only saysthat there are at least twopointsequiosillating. Using thefat that

𝜑(𝑥, 𝑦)

^has^one-sided

derivativeseverywhere,andthatforanydiretion

𝛾

^this^one-sided^derivative

𝐷 𝛾 𝜑(𝑥, 𝑦)

^must^benon-negative at

(𝑥, 𝑦 )

^we^an ^obtainequiosillationof threepoints. First wemust desribetheshapeof

𝐹 (𝑥, 𝑦, 𝑚)

^as^a

funtionof

𝑚

^.

(13)

Lemma 4.11. Let

1 ≤ 𝑥 ≤ 𝑦

^(without ^loss ^of generality). The two loal maxima

𝑚 1 (𝑥, 𝑦) ≤ 𝑚 2 (𝑥, 𝑦)

^of

𝐹 (𝑥, 𝑦, 𝑚)

^inside

𝑚 > 1

^are^given ^by

𝑚 ² ₁ (𝑥, 𝑦) = 2𝑥 + 2𝑦 + 𝐿𝑥 ² + 𝐿𝑦 ² − √ 𝑝 𝑥,𝑦 (𝐿)

2𝐿 ,

𝑚 ² ₂ (𝑥, 𝑦) = 2𝑥 + 2𝑦 + 𝐿𝑥 ² + 𝐿𝑦 ² + √ 𝑝 𝑥,𝑦 (𝐿)

2𝐿 ,

𝑝 𝑥,𝑦 (𝐿) = (𝑥 ² − 𝑦 ² ) ² 𝐿 ² + 4(𝑥 ³ − 𝑥𝑦 ² − 𝑥 ² 𝑦 + 𝑦 ³ )𝐿 +4(𝑥 + 𝑦) ² > 0.

Furthermore, if

𝑥 < 𝑦

^then

𝑚 1 (𝑥, 𝑦) ∈ (𝑥, 𝑦)

^and

𝑚 2 ∈ (𝑦, ∞ )

^. ^If

𝑥 = 𝑦

^, ^then

𝑚 1 (𝑥, 𝑥) = 𝑥

^and

𝑚 2 (𝑥, 𝑥) ∈ (𝑥, ∞ )

^.

Theritialpoints

𝑚 1

^and

𝑚 2

^also^depend^on

𝐿

^. ^When^that^dependene^must^be^made^expliit,^we^write

𝑚 1 (𝑥, 𝑦, 𝐿)

^and

𝑚 2 (𝑥, 𝑦, 𝐿)

^.

Proof. Wesetthederivative

∂𝐹/∂𝑚

^to^zero,^obtaining

0 = ∂𝐹

∂𝑚 (𝑥, 𝑦, 𝑚)

= 2 𝑥 − 𝑚 (𝑥 + 𝑚) ³

𝑦 − 𝑚

(𝑦 + 𝑚) ³ 𝑞 𝑥,𝑦 (𝑚)𝑒 ⁻ ^2𝐿𝑚 , 𝑞 𝑥,𝑦 (𝑚) = − 𝐿𝑚 ⁴ + (2𝑥 + 2𝑦 + 𝐿𝑥 ² + 𝐿𝑦 ² )𝑚 ²

− (2𝑦 ² 𝑥 − 2𝑥 ² 𝑦 − 𝐿𝑥 ² 𝑦 ² ).

The hoies

𝑚 = 𝑥

^and

𝑚 = 𝑦

^lead ^to

𝐹 (𝑥, 𝑦, 𝑚) = 0

^so ^they âre ^minima, ^so ^we ^restrit ôur âttention

to

𝑞 𝑥,𝑦 (𝑚) = 0

^. ^Bysubstituting

𝑚 ² = 𝑛

^we^obtain^a ^quadrati^polynomialⁱⁿ

𝑛

^whose ^roots ^are

𝑚 ² ₁ (𝑥, 𝑦)

and

𝑚 ² ₂ (𝑥, 𝑦)

^; ^we ^need ^to ^show ^that

1 ≤ 𝑚 1 ≤ 𝑚 2

^. ^First ^to ^show ^that ^they ^are ^real, ^we ^show ^that

the disriminant

𝑝 𝑥,𝑦 (𝐿)

^is non-negative. It sues to show that, as a quadrati polynomial over

𝐿

^, ^its

oeientsarenon-negative. Theoeientof

𝐿 ²

ând^theônstantôeientâre^squares ând^hene^non-

negative. Werewrite

𝑥 = 𝜆𝑦

^with

𝜆 ∈ (0, 1]

ând^substituteînto^theôeientôf

𝐿

ⁱⁿ

𝑝 𝑥,𝑦 (𝐿)

^to ^obtain^the

expression

𝑦 ³ (4𝜆 ³ − 4𝜆 ² − 4𝜆 + 4) > 0

^sine

𝑦 > 0

^. ^Hene

𝑝 𝑥,𝑦 (𝐿) > 0

^and

𝑚 1 < 𝑚 2

^.

Assume that

𝑥 < 𝑦

^are ^xed. ^Sine

𝐹(𝑥, 𝑦, 𝑥) = 0

^and

𝐹(𝑥, 𝑦, 𝑦) = 0

^but

𝐹 (𝑥, 𝑦, 𝑚) > 0

^for ^any

𝑚 ∈ (𝑥, 𝑦)

^, ^there ^is ^a ^loal ^maximum ⁱⁿ

𝑚

ⁱⁿ ^the ^interval

(𝑥, 𝑦)

^. ^Moreover, ^sine

𝐹 (𝑥, 𝑦, 𝑦) = 0

^and

lim 𝑚 →∞ 𝐹 (𝑥, 𝑦, 𝑚) = 0

^, ^there ^must ^be ^another ^loal ^maximum ⁱⁿ ^the^interval

(𝑦, ∞ )

^. ^However,^we^have

already shown that the onlyandidates for loal maxima are

𝑚 1

^and

𝑚 2

^, ^and ^sine

𝑚 1 ≤ 𝑚 2

^it ^must ^be

that

𝑚 1 ∈ (𝑥, 𝑦)

^and

𝑚 2 ∈ (𝑦, ∞ )

^.

Lemma4.12. Let

(𝑥 0 , 𝑦 0 )

^minimize

𝜑(𝑥, 𝑦)

^. ^Then

𝑀 (𝑥, 𝑦) = { 1, 𝑚 1 (𝑥, 𝑦), 𝑚 2 (𝑥, 𝑦) }

^with

𝑥 ∕ = 𝑦

^.

Proof. Thederivativeof

𝜑

ⁱⁿ^the^diretion

𝛾 = (𝛼, 𝛽)

^is^given^by

𝐷 𝛾 𝜑(𝑥, 𝑦) = max

𝑚 ∈ 𝑀(𝑥,𝑦) 𝛼 ∂𝐹

∂𝑥 (𝑥, 𝑦) + 𝛽 ∂𝐹

∂𝑦 (𝑥, 𝑦)

= max

𝑚 ∈ 𝑀(𝑥,𝑦) (𝛼, 𝛽) ⋅ 𝐺(𝑚),

where

𝐺(𝑚) = 4𝑚𝑒 ⁻ ^2𝐿𝑚 𝑥 − 𝑚 (𝑥 + 𝑚) ²

𝑦 − 𝑚 (𝑦 + 𝑚) ²

( 𝑦 − 𝑚 𝑥 + 𝑚 , 𝑥 − 𝑚

𝑦 + 𝑚 )

.

Inthease

𝑥 = 𝑦

^,

𝑚 1

îsâtuallyâ^minimum^so

𝑀 (𝑥, 𝑥) = { 1, 𝑚 2 (𝑥, 𝑥) }

^andⁱⁿ^partiular

𝑥 > 1

^. ^The^vetors

( 𝑥 − 1

𝑥 + 1 , 𝑥 − 1 𝑥 + 1

)

and

( 𝑥 − 𝑚 2

𝑥 + 𝑚 2 , 𝑥 − 𝑚 2

𝑥 + 𝑚 2

)

Optimized domain decomposition methods for the spherical Laplacian

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