Technische Universität Graz

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1 Technsche Unverstät Graz Challenges and Applcatons of Boundary Element Doman Decomposton Methods O. Stenbach Berchte aus dem Insttut für Numersche Mathematk Bercht 2006/9

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3 Technsche Unverstät Graz Challenges and Applcatons of Boundary Element Doman Decomposton Methods O. Stenbach Berchte aus dem Insttut für Numersche Mathematk Bercht 2006/9

4 Technsche Unverstät Graz Insttut für Numersche Mathematk Steyrergasse 30 A 8010 Graz WWW: c Alle Rechte vorbehalten. Nachdruck nur mt Genehmgung des Autors.

5 Challenges and Applcatons of Boundary Element Doman Decomposton Methods Olaf Stenbach Insttute for Computatonal Mathematcs, Graz Unversty of Technology, Steyrergasse 30, 8010 Graz, Austra, Abstract Boundary ntegral equaton methods are well suted to represent the Drchlet to Neumann maps whch are requred n the formulaton of doman decomposton methods. Based on the symmetrc representaton of the local Steklov Poncaré operators by a symmetrc Galerkn boundary element method, we descrbe a stablsed varatonal formulaton for the local Drchlet to Neumann map. By a strong couplng of the Neumann data across the nterfaces, we obtan a mxed varatonal formulaton. For borthogonal bass functons the resultng system s equvalent to nonredundant fnte and boundary element tearng and nterconnectng methods. We wll also address several open questons, deas and challengng tasks n the numercal analyss of boundary element doman decomposton methods, n the mplementaton of those algorthms, and ther applcatons. 1 Introducton Boundary element methods are well establshed approxmaton methods to solve exteror boundary value problems, or to solve partal dfferental equatons wth (pecewse) constant coeffcents consdered n complcated substructures and n domans wth movng boundares. For an state of the art overvew on recent advances on mathematcal aspects and engneerng applcatons of boundary ntegral equaton methods, see, for example, Schanz and Stenbach [2007]. However, for partal dfferental equatons wth nonlnear coeffcents the couplng of fnte and boundary element methods seems to be an effcent tool to solve complex problems n complcated domans. For the formulaton and for an effcent soluton of the resultng systems of equatons, doman decomposton methods are mandatory. The classcal approach to couple fnte and boundary element methods s to use only the weakly sngular boundary ntegral equaton wth sngle and double layer potentals, see, e.g., Brezz and Johnson [1979], Johnson and Nedelec [1980], and Wendland [1988]. In Costabel [1987] a symmetrc couplng of fnte and boundary elements usng the so called hypersngular boundary ntegral operator was ntroduced. Ths approach was then extended to symmetrc Galerkn boundary element methods, see, e.g., Hsao and Wendland [1990]. Approprate precondtoned teratve strateges were later consdered n Carstensen et al. [1998], whle qute general precondtoners based on operators of the opposte order were ntroduced n Stenbach and Wendland [1998]. Boundary element tearng and nterconnectng (BETI) methods were descrbed n Langer and Stenbach [2003] as counterpart of FETI methods whle n Langer et al. [2006] these methods were combned wth a fast multpole approxmaton of the local boundary ntegral operators nvolved. For an alternatve approach to boundary ntegral doman decomposton methods see also Khoromskj and Wttum [2004]. Here we wll gve a qute general settng of tearng and nterconnectng, or more general, hybrd doman decomposton methods. The local partal dfferental equaton s rewrtten as a local Drchlet to Neumann map whch can be realzed ether by doman varatonal formulatons or by usng boundary ntegral formulatons. Snce the related functon spaces are fractonal Sobolev spaces, one may ask for the rght defnton of the assocated norms. It turns out that the used norms whch are nduced by the local sngle layer potental or ts nverse allows for almost explcte spectral equvalence nequaltes, and an approprate stablsaton of the sngular Steklov Poncaré operators. The modfed Drchlet to Neumann map s then used to obtan a mxed varatonal formulaton allowng a weak couplng of the local Drchlet data. However, stayng wth a globally conform method and usng borthogonal bass functons we end up wth the standard tearng and nterconnectng approach as n FETI and n BETI. 5

6 The am of ths paper s to sketch some deas to obtan advanced formulatons n boundary ntegral doman decomposton methods, to propose to use specal norms n the numercal analyss, and to state some challengng tasks n the mplementaton of fast boundary element doman decomposton algorthms to solve challengng problems from engneerng and ndustry. 2 Boundary Integral Equaton DD Methods As a model problem we consder the Drchlet boundary value problem of the potental equaton, dv[α(x) u(x)] = f(x) for x Ω, u(x) = g(x) for x Γ (1) where Ω R 3 s a bounded doman wth Lpschtz boundary Γ = Ω. We assume that there s gven a non overlappng doman decomposton Ω = p Ω, Ω Ω j = for j, Γ = Ω. (2) =1 The doman decomposton as consdered n (2) may arse from a pecewse constant coeffcent functon α(x) due to the physcal model, n partcular we may assume α(x) = α for x Ω. However, to construct effcent soluton strateges n parallel, one may also ntroduce a doman decomposton (2) when consderng the Laplace or Posson equaton n a complcated three dmensonal structure. A challengng task s to fnd a doman decomposton (2) whch s based on boundary nformatons only,.e., wthout any addtonal volume meshes. Usng deas as used n fast boundary element methods,.e. by a bsecton algorthm t s possble to decompose a gven boundary mesh nto two separate surface meshes. Whle ths step seems to be smple, the delcate task s the defnton of the new nterface mesh whch should take care of the gven geometrc stuaton,.e. one should avod the ntersecton of the new nterface wth the orgnal boundary. We have already appled ths algorthm to fnd a sutable doman decomposton of the Lake St. Wolfgang doman as shown n Fgure 1. Fgure 1: Doman Decomposton of the Lake St. Wolfgang Doman. It seems to be an open problem to fnd and to mplement a robust algorthm for an automatc doman decomposton of complcated three dmensonal strutures whch s based on surface nformatons only. Such a tool s essentally needed when consderng boundary element doman decomposton methods. Prelmnary results on ths topc wll be publshed elsewhere (Of and Stenbach [2007]). A smlar approach was already used n Ivanov et al. [2006] to desgn an automatc doman decomposton approach for unstructured grds n three dmensons. There, the remeshng of the new nterface s done wthn the splttng hyperplane wthout consderng the robustness of the algorthm for complcated geometres. Instead of the global boundary value problem (1) we now consder the local boundary value problems α u (x) = f (x) for x Ω, u (x) = g(x) for x Γ Γ (3) 6

7 together wth the transmsson boundary condtons u (x) = u j (x), α t (x) + α j t j (x) = 0 for x Γ j = Γ Γ j, (4) where t = n u s the exteror normal dervatve of u on Γ. Snce the soluton u of the local boundary value problem (3) s gven va the representaton formula u (x) = 1 t (y) 4π x y ds y 1 1 u (y) 4π n y x y ds y f (y) α 4π x y dy Γ Γ for x Ω, t s suffcent to fnd the complete Cauchy data [u, t ] Γ whch are related to the solutons u of the local boundary value problems (3). The approprate boundary ntegral equatons to derve a boundary ntegral representaton of the nvolved Drchlet to Neumann map are gven by means of the Calderon projector ( ) ( 1 u = 2 I K ) ( ) V u 1 t D 2 I ( ) N0 f, K t α N 1 f where V s the sngle layer potental, K s the double layer potental, D s the hypersngular boundary ntegral operator, and N j f are some Newton potentals, respectvely. Hence, we fnd the Drchlet to Neumann map as wth the Steklov Poncaré operator α t (x) = α (S u )(x) (N f )(x) for x Γ (5) (S u )(x) = V 1 ( 1 2 I + K )u (x) (6) [ = D + ( 1 2 I + K )V 1 ( 1 ] 2 I + K ) u (x) for x Γ. (7) Note that N f = V 1 N 0 f. Replacng the partal dfferental equaton n (3) by the related Drchlet to Neumann map (5) ths results n a coupled formulaton to fnd the local Cauchy data [u, t ] Γ such that Ω α t (x) = α (S u )(x) (N f )(x) for x Γ, u (x) = g(x) for x Γ Γ, u (x) = u j (x) for x Γ j, α t (x) + α j t j (x) = 0 for x Γ j. (8) In what follows we frst have to analyze the local Steklov Poncare operators S : H 1/2 (Γ ) H 1/2 (Γ ). Snce we are dealng wth fractonal Sobolev spaces H ±1/2 (Γ ) one may ask for approprate norms to be used. It turns out that norms whch are nduced by the local sngle layer potentals V may be advantageous. In partcular, V = V, Γ, V 1 = V 1, Γ are equvalent norms n H 1/2 (Γ ) and n H 1/2 (Γ ), respectvely. Wth the contracton property of the double layer potental (Stenbach and Wendland [2001]), ( 1 2 I + K )v V 1 c K, v V 1 for all v H 1/2 (Γ ) (9) where the constant c K, = cd 1 cv 1 < 1 7

8 s only shape senstve, we have S v V = ( 1 2 I + K )v V 1 c K, v V 1 for all v H 1/2 (Γ ) as well as S v, v Γ (1 c K, ) v 2 for all v V 1 H 1/2 (Γ ), v 1. In partcular, the constants form the non trval kernel of the local Steklov Poncare operators S,.e., S 1 = 0 n the sense of H 1/2 (Γ ). To realze the related orthogonal splttng of H 1/2 (Γ ) we ntroduce the natural densty w eq, H 1/2 (Γ ) as the unque soluton of the local boundary ntegral equaton V w eq, = 1. Then we may defne the stablzed hypersngular boundary ntegral operator S : H 1/2 (Γ ) H 1/2 (Γ ) va the Resz representaton theorem by the blnear form S u, v Γ = S u, v Γ + β u, w eq, Γ v, w eq, Γ, β R +. (10) Theorem 2.1 Let S : H 1/2 (Γ ) H 1/2 (Γ ) be the stablzed Steklov Poncaré operator as defned n (10). Then there hold the spectral equvalence nequaltes c e S 1 V 1 for all v H 1/2 (Γ ) wth postve constants S v, v Γ S v, v Γ c e 2 1 V v, v Γ c e S 1 = mn{1 c K,, β 1, w eq, Γ }, c e S 2 = max{c K,, β 1, w eq, Γ }. Therefore, an optmal scalng s gven for β = 1 2 1, w eq, Γ, c e S 1 = 1 c K,, c e S 2 = c K,. Hence, the Drchlet to Neumann map (5) can be wrtten n a modfed varatonal formulaton as α t, v Γ = S ũ, v Γ N f, v Γ for all v H 1/2 (Γ ) (11) when assumng the local solvablty condtons α t, 1 Γ + N f, 1 Γ = 0. (12) In partcular, nsertng v = 1 nto the modfed Drchlet to Neumann map (11), we obtan from the solvablty condton (12) 0 = α t, 1 Γ + N f, 1 Γ = S ũ, 1 Γ + β ũ, w eq, Γ 1, w eq, Γ and therefore the scalng condton due to ũ, w eq, Γ = 0 (13) S ũ, 1 Γ = ũ, S 1 Γ = 0, 1, w eq, Γ = 1, V 1 1 Γ > 0. In fact, the scalng condton (13) s the natural characterzaton of functons ũ H 1/2 (Γ ) whch are orthogonal to the constants n the sense of H 1/2 (Γ ). Hence, the local Drchlet datum s gven va u = ũ + γ, γ R. Now, the coupled formulaton (8) can be rewrtten as α t (x) = α ( S ũ )(x) (N f )(x) for x Γ, ũ (x) + γ = g(x) for x Γ Γ, ũ (x) + γ = ũ j (x) + γ j for x Γ j, α t (x) + α j t j (x) = 0 for x Γ j, α t, 1 Γ + N f, 1 Γ = 0 (14) 8

9 where we have to fnd ũ H 1/2 (Γ ), t H 1/2 (Γ ), and γ R, = 1,...,p. Hereby, the varatonal formulaton of the modfed Drchlet to Neumann map reads: Fnd ũ H 1/2 (Γ ) such that α S ũ, v Γ α t, v Γ = N f, v Γ (15) s satsfed for all v H 1/2 (Γ ), = 1,...,p. The Neumann transmsson condtons n weak form are α t + α j t j, v j Γj = [α t (x) + α j t j (x)]v j (x)ds x = 0 (16) Γ j for all v j H 1/2 (Γ j ). Takng the sum over all nterfaces Γ j, ths s equvalent to p α t, v Γ Γ\Γ = 0 for all v H 1/2 (Γ S ), (17) =1 where Γ S = p =1 Γ s the skeleton of the gven doman decomposton. The Drchlet transmsson condtons n (14) can be wrtten as [ũ + γ ] [ũ j + γ j ], τ j Γj = 0 for all τ j H 1/2 (Γ j ) = [H 1/2 (Γ j )], (18) whle the Drchlet boundary condtons n weak form read ũ + γ, τ 0 Γ Γ = g, τ 0 Γ Γ for all τ 0 H 1/2 (Γ Γ). (19) In addton we need to have the local solvablty condtons α t, 1 Γ + N f, 1 Γ = 0. (20) The coupled varatonal formulaton (15) (20) s n fact a mxed (saddle pont) doman decomposton formulaton of the orgnal boundary value problem (1). Hence we have to ensure a certan stablty (BBL) condton to be satsfed,.e., a stable dualty parng between the prmal varables ũ and the dual Lagrange varable t along the nterfaces Γ j. Note that we also have to ncorporate the addtonal constrants (20) and ther assocated Lagrange multplers γ. Whle the unque solvablty of the contnuous varatonal formulaton (15) (20) follows n a qute standard way, as, e.g. n Stenbach [2003], the stablty of an assocated dscrete scheme s not so obvous. Clearly, the Galerkn dscretzaton of the coupled problem (15) (20) depends on the local tral spaces to approxmate the local Cauchy data [ũ, t ]. In partcular, the varatonal formulaton (15) (20) may serve as a startng pont for Mortar doman decomposton or three feld formulatons as well (see Stenbach [2003] and the references gven theren). However, here we wll consder only an approach whch s globally conform. Let Sh 1(Γ S) be a sutable tral space on the skeleton Γ S, e.g., of pecewse lnear bass functons ϕ k, k = 1,...,M, and let Sh 1(Γ ) denote ts restrcton onto Γ, where the local bass functons are ϕ k, k = 1,...,M. In partcular, A R M M are connectvty matrces lnkng the global degrees of freedom u R M u h Sh 1(Γ S) to the local ones, u = A u R M u h Γ Sh 1(Γ ). Moreover, let Sh 0(Γ j) be some tral space to approxmate the local Neumann data t and t j along the nterface Γ j, for example we may use pecewse constant bass functons ψs j. In the same way we ntroduce bass functons ψs 0 S0 h (Γ) to approxmate the Neumann data along the Drchlet boundary Γ. The tral spaces Sh 0(Γ j) and Sh 0(Γ) defne a global tral space S0 h (Γ S) of pecewse constant bass functons ψ ι mplyng λ h Sh 0(Γ S) λ R N,.e., we have λ h Γj Sh 0(Γ j) λ j R Nj and λ h Γ Sh 0(Γ) λ 0 R N0. For the global tral space S 0 h (Γ S) = <j S 0 h (Γ j) S 0 h (Γ) = span{ψ ι} N ι=0, we defne the restrctons ψs j = rj ι ψ ι wth rι j = 1, rι j x Γ. Hence we can also ntroduce a local mappng = 1 for < j, and ψ 0 s = r0 ι ψ ι, r 0 ι = 1 for t = 1 α R λ R N for λ R N 9

10 satsfyng R [s, ι] = rι j = 1, R j [s j, ι] = rι j = 1 for ι = 1,...,N, s = 1,...,N, < j, and R [s, ι] = r 0 ι = 1 for x Γ. For the assocated approxmatons t,h S 0 h (Γ ) t R N, we then fnd α t,h (x) + α j t j,h (x) = 0 for x Γ j,.e., the Neumann transmsson condtons (16) are satsfed n a strong sense. The Galerkn approxmaton of the Drchlet transmsson condton (18) can now be wrtten as [ M ] M ũ,k ϕ k (x) + γ j ũ j,k ϕ j k (x) + γ j ψj σ (x)ds x = 0 Γ j Γ j k=1 k=1 for σ = 1,...,N j, and < j, or for ι = 1,...,N [ M ] M ũ,k ϕ k (x) + γ rι j ψ j ι(x) + ũ j,k ϕ j k (x) + γ j rι j ψ ι(x)ds x = 0. k=1 Correspondngly, the Galerkn dscretzaton of the Drchlet boundary condton (19) reads Γ Γ [ M k=1 ũ,k ϕ k(x) + γ ]r ι 0 ψ ι (x)ds x = k=1 Γ Γ g(x)r 0 ι ψ ι (x)ds x. Combnng both the Galerkn dscretzaton of the Drchlet transmsson and of the Drchlet boundary condtons, we can wrte where B R M M are defned by B [ι, k] = ϕ k(x)r j ι ψ ι (x)ds x, B [ι, k] = Γ j p B ũ + Gγ = g (21) =1 Γ Γ ϕ k(x)r 0 ι ψ ι (x)ds x. In addton, the matrx G = (G 1,..., G p ) R M p and the vector g R M of the rght hand sde are defned correspondngly,.e. G [ι, ] = r j ι ψ ι (x)ds x, G [ι, ] = rι 0 ψ ι(x)ds x. Γ j Γ Γ In partcular, we have G = B 1 where 1 R M s the coeffcent vector whch s related to the constant functon 1 H 1/2 (Γ ). Moreover, from the solvablty condtons (20) we obtan G λ = q = N f, 1 Γ for = 1,...,p. The Galerkn dscretzaton of the local Drchlet to Neumann map (15) fnally gves α S,h ũ B λ = f, where we have to approxmate the exact stffness matrx S,h ncludng the local Steklov Poncaré operator S, e.g., n the symmetrc representaton (7), by some boundary element dscretzaton, S,h = D,h + ( 1 2 M,h + K,h)V 1,h (1 2 M,h + K,h ) + β a a, 10

11 where the local boundary element matrces are gven as D,h [l, k] = D ϕ k, ϕ l Γ, K,h [ν, k] = K ϕ k, ϑ ν Γ, V,h [ν, µ] = V ϑ µ, ϑ ν Γ, M,h [ν, k] = ϕ k, ϑ ν Γ, a,k = ϕ k, w eq, Γ for k, l = 1,...,M, µ, ν = 1,..., N where span{ϑ µ } N µ=1 H 1/2 (Γ ) s some local boundary element tral space to approxmate the local Neumann data whch are needed n the defnton of the approxmate Steklov Poncaré operator. Note that the bass functons ϑ µ can be defned n an almost arbtrary way, we only have to assume some approxmaton property to ensure convergence of the dscrete scheme. The most smplest choce would be to dentfy the bass functons ϑ µ whch are defned along the skeleton. In an analogue manner, one may even defne an approxmate Steklov Poncaré operator by usng local fnte elements, see, e.g., Langer and Stenbach [2004]. Summarzng the above, we end up wth a global system of lnear equatons, wth ψ j s α 1 S1,h B α p Sp,h B p B 1 B p G G ũ 1. ũ p = λ γ f 1. f. (22) p g q The unque solvablty of the lnear system (22) and therefore of the coupled varatonal problem (15) (20) follows from some stablty (LBB) condton lnkng the local tral spaces Sh 1(Γ ) and Sh 0(Γ j) along the couplng nterface Γ j. Here, we only consder the specal case where the bass functons ϕ k and ψj s are borthogonal,.e. { ϕ k(x)ψ j 1 for s = k, s (x)ds x = 0 for s k. Γ j Then, the entres of the matrces B consst just of zeros and ±1 descrbng a nodal couplng of the assocated prmal degrees of freedom. In partcular, the use of borthogonal bass functons to dscretze the coupled varatonal problem (15) (20) s equvalent to a redundant fnte or boundary element tearng and nterconnectng approach for a standard doman decomposton formulaton, see, e.g., Langer and Stenbach [2004]. Whle for matchng grds the descrbed formulaton s a conform dscretzaton scheme, t may be generalzed to dfferent local grds and dfferent local tral spaces as well. Ths leads mmedately to hybrd or mortar doman decomposton methods where the choce of local tral spaces s essental to ensure the local stablty condtons, see, e.g., Wohlmuth [2001] and the references gven theren. Snce the approxmaton of the local Drchlet to Neumann maps can be done by any avalable dscrezaton scheme, the presented formulaton allows the couplng of dfferent dscretzaton schemes such as fnte and boundary element methods, and the couplng of locally dfferent meshes and tral spaces. However, when consderng a boundary element approxmaton of the Steklov Poncaré operator S u = [D + ( 1 2 I + K )V 1 ( 1 2 I + K )]u = D u + ( 1 2 I + K )w the local Neumann data w = V 1 ( 1 2 I + K ) concde wth the Neumann data t as used n the coupled formulaton (14). It seems to be an open problem how ths relaton can be used to fnd further advanced boundary element doman decomposton formulatons, n partcular to fnd more effcent precondtoned teratve strateges to solve the resultng lnear systems of equatons n parallel. 11

12 3 Conclusons For the numercal analyss of standard boundary element methods see, for example, Stenbach [2007]. Snce the dscretzaton of non local boundary ntegral operators wth sngular kernel functons leads to dense stffness matrces, the use of fast boundary element methods s an ssue. For an overvew of those methods, and for the mplementaton and for the applcaton of the Adaptve Cross Approxmaton approach, see Rjasanow and Stenbach [2007]. Other possble fast boundary element methods are the Fast Multpole Method, see, e.g., Of et al. [2006] and the references theren, or Herarchcal Matrces, see, e.g., Hackbusch et al. [2005]. The teratve soluton of the lnear system (22) of the boundary element tearng and nterconnectng approach can be done by a projected precondtoned conjugate gradent method n a specal nner product snce the system matrx has a two fold saddle pont structure, see also Langer et al. [2006], where we have also descrbe approprate precondtonng strateges. Snce the potental equaton (1) s just a model problem, the methodology gven n ths paper can be extended to more advanced problems n a straght forward way, e.g., for problems n lnear elastostatcs, for almost ncombressble materals and for the Stokes problem. More challengng are the handlng of the Helmholtz equaton or of the Maxwell system where more advanced formulatons are needed to obtan boundary ntegral equatons whch are unque solvable for all wave numbers. Acknowledgement: Ths work has been partally supported by the German Research Foundaton (DFG) under the Grant SFB 404 Multfeld Problems n Contnuum Mechancs. References F. Brezz and C. Johnson. On the couplng of boundary ntegral and fnte element methods. Calcolo, 16: , C. Carstensen, M. Kuhn, and U. Langer. Fast parallel solvers for symmetrc boundary element doman decomposton methods. Numer. Math., 79: , M. Costabel. Symmetrc methods for the couplng of fnte elements and boundary elements. In C. A. Brebba, G. Kuhn, and W. L. Wendland, edtors, Boundary Elements IX, pages , Berln, Sprnger. W. Hackbusch, B. N. Khoromskj, and R. Kremann. Drect Schur complement method by doman decomposton based on H matrx approxmaton. Comput. Vs. Sc., 8: , G. C. Hsao and W. L. Wendland. Doman decomposton methods n boundary element methods. In R. Glownsk and et. al., edtors, Doman Decomposton Methods for Partal Dfferental Equatons. Proceedngs of the Fourth Internatonal Conference on Doman Decomposton Methods, pages 41 49, Baltmore, SIAM. E. G. Ivanov, H. Andrä, and A. N. Kudryavtsev. Doman decomposton approach for automatc parallel generaton of 3d unstructured grds. In P. Wesselng, E. Onate, and J. Peraux, edtors, Proceedngs of the European Conference on Computatonal Flud Dynamcs ECCOMAS CFD 2006, TU Delft, The Netherlands, C. Johnson and J. C. Nedelec. On couplng of boundary ntegral and fnte element methods. Math. Comp., 35: , B. N. Khoromskj and G Wttum. Numercal Soluton of Ellptc Dfferental Equatons by Reducton to the Interface, volume 36 of Lecture Notes n Computatonal Scence and Engneerng. Sprnger, Berln, U. Langer and O. Stenbach. Boundary element tearng and nterconnectng methods. Computng, 71: ,

13 U. Langer and O. Stenbach. Coupled boundary and fnte element tearng and nterconnectng methods. In R. Kornhuber, R. Hoppe, J. Peraux, O. Pronneau, O. Wdlund, and J. Xu, edtors, Doman Decomposton Methods n Scence and Engneerng, volume 40 of Lecture Notes n Computatonal Scence and Engneerng, pages Sprnger, Hedelberg, U. Langer, G. Of, O. Stenbach, and W. Zulehner. Inexact data sparse boundary element tearng and nterconnectng methods. SIAM J. Sc. Comput., Accepted for publcaton. G. Of and O. Stenbach. Automatc generaton of boundary element doman decomposton meshes. In preparaton, G. Of, O. Stenbach, and W. L. Wendland. Boundary element tearng and nterconnectng methods. In R. Helmg, A. Melke, and B. I. Wohlmuth, edtors, Multfeld Problems n Sold and Flud Mechancs, volume 28 of Lecture Notes n Appled and Computatonal Mechancs, pages Sprnger, Hedelberg, S. Rjasanow and O. Stenbach. The Fast Soluton of Boundary Integral Equatons. Mathematcal and Analytcal Technques wth Applcatons to Engneerng. Sprnger, New York, M. Schanz and O. Stenbach, edtors. Boundary Element Analyss: Mathematcal Aspects and Applcatons, volume 29 of Lecture Notes n Appled and Computatonal Mechancs. Sprnger, Hedelberg, O. Stenbach. Stablty estmates for hybrd coupled doman decomposton methods, volume 1809 of Lecture Notes n Mathematcs. Sprnger, Hedelberg, O. Stenbach. Numercal Approxmaton Methods for Ellptc Boundary Value Problems. Fnte and Boundary Elements. Texts n Appled Mathematcs. Sprnger, New York, O. Stenbach and W. L. Wendland. The constructon of some effcent precondtoners n the boundary element method. Adv. Comput. Math., 9: , O. Stenbach and W. L. Wendland. On C. Neumann s method for second order ellptc systems n domans wth non smooth boundares. J. Math. Anal. Appl., 262: , W. L. Wendland. On asymptotc error estmates for combned BEM and FEM. In E. Sten and W. L. Wendland, edtors, Fnte Element and Boundary Element Technques from Mathematcal and Engneerng Pont of Vew, volume 301 of CISM Courses and Lectures, pages Sprnger, Wen, New York, B. I. Wohlmuth. Dscretzaton Methods and Iteratve Solvers Based on Doman Decomposton, volume 17 of Lecture Notes n Computatonal Scence and Engneerng. Sprnger, Berln,

14 Erschenene Preprnts ab Nummer 2005/1 2005/1 O. Stenbach Numersche Mathematk 1. Vorlesungsskrpt. 2005/2 O. Stenbach Technsche Numerk. Vorlesungsskrpt. 2005/3 U. Langer Inexact Fast Multpole Boundary Element Tearng and Interconnectng G. Of Methods O. Stenbach W. Zulehner 2005/4 U. Langer Inexact Data Sparse Boundary Element Tearng and Interconnectng G. Of Methods O. Stenbach W. Zulehner 2005/5 U. Langer Fast Boundary Element Methods n Industral Applcatons O. Stenbach Söllerhaus Workshop, , Book of Abstracts. W. L. Wendland 2005/6 U. Langer Dual Prmal Boundary Element Tearng and Interconnectng Methods A. Pohoata O. Stenbach 2005/7 O. Stenbach (ed.) Jahresbercht 2004/ /1 S. Engleder Modfed Boundary Integral Formulatons for the O. Stenbach Helmholtz Equaton. 2006/2 O. Stenbach 2nd Austran Numercal Analyss Day. Book of Abstracts. 2006/3 B. Muth Collson Detecton for Complcated Polyhedra Usng G. Of the Fast Multpole Method of Ray Crossng P. Eberhard O. Stenbach 2006/4 G. Of Numercal Tests for the Recovery of the Gravty Feld B. Schneder by Fast Boundary Element Methods 2006/5 U. Langer 4th Workshop on Fast Boundary Element Methods n Industral O. Stenbach Applcatons. Book of Abstracts. W. L. Wendland 2006/6 O. Stenbach (ed.) Jahresbercht 2005/ /7 G. Of The All floatng BETI Method: Numercal Results 2006/8 P. Urthaler Automatsche Postonerung von FEM Netzen G. Of O. Stenbach