[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-228125-en":3,"doc-seo-228125-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":26,"seo_description":14,"update_tm":27,"read_time":28},228125,2336477405376,"Stanley","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Optimized Schwarz Methods for Domains with an Arbitrary Interface","Optimized Schwarz methods accelerate domain decomposition for partial differential equations by applying first- or higher-order boundary conditions on an artificial interface. The interface operator is expressed using Poincaré–Steklov operators to bound the spectral radius for Poisson-like problems on two nonoverlapping, essentially arbitrary subdomains. For Robin boundary operators, an optimal parameter choice yields an upper bound with rate 1 − O(h^{1/2}); for a higher-order operator, a two-parameter choice yields 1 − O(h^{1/4}). These estimates match predicted rectangular-subdomain behavior and numerical simulations.","Optimized Schwarz Methods for Domains with an Arbitrary Interface  \nShiu Hong Lui  \nDepartment of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2, [luish@cc.umanitoba.ca](luish@cc.umanitoba.ca)  \n1 Introduction  \nOptimized Schwarz methods form a class of domain decomposition methods for the solution of partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate its convergence. In the literature, analysis of optimized Schwarz methods rely on Fourier analysis and so the domains are restricted to be regular (rectangle or disk) . By expressing the interface operator in terms of Poincare–Steklov operators, we are able to derive upper bounds of the spectral radius of the operator for Poisson-like problems for two essentially arbitrary subdomains. For a first order (Robin) boundary operator, an optimal choice of the parameter in the boundary operator leads to an upper bound of 1 − O (h1/2) of the spectral radius, where h is the discretization parameter. For a certain higher order boundary operator, a clever choice of the two parameters in the boundary operator leads to an upper bound of 1 − O (h 1/4) of the spectral radius. These agree with the predicted rates for rectangular subdomains available in the literature and are also the observed rates in numerical simulations. This contribution summarizes the author’s work in [ 11, 12] .  \nLet Ω be a bounded domain in IRN with a smooth boundary. Suppose Ω is composed of two nonoverlapping open subdomains, that is, Ω = Ω 1 ∪ Ω 2 with Ω1 ∩ Ω2 = ∅ . Assume that the artificial boundary Γ = Ω 1 ∩ Ω 2 is non-trivial (nonzero measure in RN −1) and is a smooth curve. We shall always assume that ∂Ωi \\ Γ is non-trivial for both i = 1 , 2.  \nRecall the trace space  \nH10/02 (Γ ) = {v| Γ , v ∈ H10( Ω )}  \nwith dual H −1/2(Γ ) . For i = 1 , 2, let  \nVi = {vi ∈ H1 ( Ωi ) , vi = 0 on ∂Ωi ∩ ∂Ω} .  \nDefine the trace operators Ti : Vi → H10/02 (Γ ) by  \n110 S.H. Lui  \nTivi = vi | Γ , vi ∈ Vi.  \nFor simplicity, consider the model problem  \n−􀀞u = f on Ω, u = 0 on ∂Ω .  \nOne candidate for the subdomain problem is  \n−􀀞ui = f on Ωi ,  \nui = p on Γ  \nwith ui ∈ Vi for some function p ∈ H10/02 (Γ ) . Note that p is the correct function (p = Tiu) if  \nuν11 + uν22 = 0 on Γ.  \nThis is known as the transmission condition. Define u i = uei +zi where uei = Hip ∈ Vi is the harmonic extension of p:  \n−􀀞uei = 0 on Ωi , uei = p on Γ  \nand zi = 􀀞 −i1 f where 􀀞 i is the Laplacian operator with domain H10( Ωi ) . Define the Poincare–Steklov operators Si : H10/02 (Γ ) → H −1/2(Γ ) by  \nSip = ∂iipor by  \n􀀒Sip, q􀀓 = 􀀔Ωi ∇pe · ∇qe , ∀p, q ∈ H10/02 (Γ )  \nwith pe = Hip, qe = Hi q. In the above inner product, Si is self-adjoint and positive definite. Hence the transmission condition can also be expressed as  \n(S1 + S2 )u| Γ = w (1)  \nfor some w.  \n2 First-Order Boundary Condition  \nIn [10], the author defined the Schwarz sequence {ui(n) ∈ Vi , n ≥ 0} by  \n−􀀞ui(n) = f on Ωi ,  \n∂~~i~~ν(ni) + λui(n) = gi(n) on Γ. (2)  \nOptimized Schwarz Methods for Domains with an Arbitrary Interface 111  \nHere λ is a positive constant. Noting that ν 1 = −ν2 on Γ , the Robin data can be updated as  \ng3(+i1) = − ∂~~i~~ν(ni) + λui(n) on Γ, i = 1 , 2. The iteration can be started for any initial gi(0) ∈ L2 (Γ ) . In practice, the choice gi(0) = 0 is convenient.  \nThe following is an equivalent update ([2]):  \ng3(+i1) = 2λui(n) − gi(n) on Γ, i = 1 , 2. (3)  \nNote that the subdomain computations can be carried out concurrently. Many authors have studied the convergence of this method and the choice of the optimal parameter. See [1, 12, 15] which are most pertinent to this paper.  \nThe function g2 can be eliminated in (3) to obtain the following equation for g1 : 1I − (I − 2λ(S2 + λ)−1)(I − 2λ(S1 + λ)−1)2g 1  \n= 2λb ≡ 2λ(T2 z2 − (I − 2λ(S2 + λ)−1)T1 z 1 ) . The operator for g 1 has alternative representations  \nI − (S2 + λ)−1(S2 − λ)(S1 − λ)(S1 + λ)−1 ","cbCaikrfEn7mA9e7","https://ap.wps.com/l/cbCaikrfEn7mA9e7","pdf",145415,1,"English","en",105,"# Introduction\n## First-Order Boundary Condition\n## Poincaré–Steklov Operators and Transmission Condition","[{\"question\":\"What makes optimized Schwarz methods converge faster in this work?\",\"answer\":\"They employ first- or higher-order boundary conditions on the artificial interface to accelerate convergence of the domain decomposition iteration.\"},{\"question\":\"How is the interface operator analyzed?\",\"answer\":\"The interface operator is represented via Poincaré–Steklov operators, enabling upper bounds on the spectral radius for Poisson-like problems on two arbitrary subdomains.\"},{\"question\":\"What convergence rates are obtained for different boundary operators?\",\"answer\":\"For a first-order (Robin) boundary operator, an optimal parameter choice gives an upper bound of 1 − O(h^{1/2}); for a certain higher-order operator, a suitable two-parameter choice gives 1 − O(h^{1/4}).\"}]","Optimized Schwarz Methods for Domains with an Arbitrary Interface | PDF",1789021226,20,{"code":4,"msg":30,"data":31},"ok",{"site_id":23,"language":22,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"optimized-schwarz-methods-for-domains-with-an-arbitrary-interface","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/optimized-schwarz-methods-for-domains-with-an-arbitrary-interface/228125/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":22,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-09-11","2026-09-10",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What makes optimized Schwarz methods converge faster in this work?","Question",{"text":75,"@type":76},"They employ first- or higher-order boundary conditions on the artificial interface to accelerate convergence of the domain decomposition iteration.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the interface operator analyzed?",{"text":80,"@type":76},"The interface operator is represented via Poincaré–Steklov operators, enabling upper bounds on the spectral radius for Poisson-like problems on two arbitrary subdomains.",{"name":82,"@type":73,"acceptedAnswer":83},"What convergence rates are obtained for different boundary operators?",{"text":84,"@type":76},"For a first-order (Robin) boundary operator, an optimal parameter choice gives an upper bound of 1 − O(h^{1/2}); 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