Error estimates of the DtN finite element method for the exterior Helmholtz problem
|
|
- George Little
- 5 years ago
- Views:
Transcription
1 Journal of Computational and Applied Mathematics (7) Error estimates of the DtN finite element method for the exterior Helmholtz problem Daisuke Koyama Department of Computer Science, The University of Electro-Communications, Chofugaoka 1 5 1, Chofu, Tokyo , Japan Received 13 September 4; received in revised form 7 December 5 Abstract A priori error estimates are established for the DtN (Dirichlet-to-Neumann) finite element method applied to the exterior Helmholtz problem. The error estimates include the effect of truncation of the DtN boundary condition as well as that of the finite element discretization. A property of the Hankel functions which plays an important role in the proof of the error estimates is introduced. 5 Elsevier B.V. All rights reserved. Keywords: Exterior Helmholtz problem; Artificial boundary condition; Drichlet-to-Neumann operator; Finite element method; A priori error estimate 1. Introduction We consider the exterior Helmholtz problem: Δu k u = f in Ω, u = on γ, ( ) u lim r + r(d 1)/ r iku = (the outgoing radiation condition), where k, called the wave number, is a positive constant, Ω is an unbounded domain of R d (d = or 3) with sufficiently smooth boundary γ, f is a given datum, r = x for x R d, and i = 1. Assume that O R d \Ω is a bounded open set and that f has a compact support. Problem (1) arises in models of acoustic scattering by a sound-soft obstacle O embedded in a homogeneous medium. To solve numerically problem (1), one often introduces an artificial boundary in order to reduce the computational domain to a bounded domain and imposes an artificial boundary condition on the artificial boundary. Although a variety of artificial boundary conditions have been proposed (see, e.g., [7], for a review), we focus on the exact nonlocal boundary condition based on the Dirichlet-to-Neumann (DtN) operator, which is called the exact DtN boundary condition. The comparison of the exact DtN boundary condition with local artificial boundary conditions is described in [8,1]. Imposing the exact DtN boundary condition on the artificial boundary, we can reduce problem (1) equivalently to a problem on the bounded domain between the artificial boundary and the boundary γ. We discretize the reduced problem by using the finite element method. This method is called the DtN finite element method. (1) Tel.: ; fax: address: koyama@im.uec.ac.jp /$ - see front matter 5 Elsevier B.V. All rights reserved. doi:1.116/j.cam.5.1.4
2 D. Koyama / Journal of Computational and Applied Mathematics (7) 1 31 The DtN finite element method for the exterior Helmholtz problem has first been proposed by MacCamy Marin [3] in 198, who represented the DtN operator through an integral equation. Feng [4], Masmoudi [4], and Keller Givoli [18] have derived the Fourier series representation of the DtN operator. Masmoudi [4] and Keller Givoli [18] incorporated such a representation directly into the finite element method. The DtN finite element method for other kinds of problems concerning the Helmholtz equation has been investigated by several authors. In 1978, Fix Marin [6], who are pioneers in the DtN finite element method, studied the under-water acoustic problem. Goldstein [11] established error estimates for the Helmholtz problem on unbounded waveguides. Bao [1] also established error estimates for the problem concerning the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material. In China, independently of the western world, the DtN finite element method has been suggested and developed first by Feng and Yu in 198 and 198, for example, see [5], where they called it the canonical boundary element method or the natural boundary element method. Yu has published many papers [31,3,3,35,17] and a monograph [33,34] (see also its review [9]) in this direction. In Ushijima s paper [8], we can find these words: Chinese scholars, Feng Kang, Han Houde, Yu De-hao and others should be mentioned among founders of the treatment. In western world, J.B. Keller and D. Givoli are also should be quoted. In this paper, we establish a priori error estimates in the H 1 - and L -norms for the exterior Helmholtz problem. The use of the Fourier series representation of the DtN operator requires truncating the series in practical computations. So we analyze the series truncation error as well as the finite element discretization error. To the best knowledge of the author, our error estimates are new, because no error estimate treating both the truncation error and the discretization error simultaneously has been published yet (cf. [1, p. 3]). MacCamy Marin [3] and Masmoudi [4] have derived an error estimate, but their estimate depends only on the mesh size (see also [3,17]). The counterpart of our error estimates for the Helmholtz problem on unbounded waveguides has been established by Goldstein [11] in 198. Our error analysis roughly follows his analysis; however, we need some properties of the Hankel functions, which contain a new and important result (Lemma 4); we were inspired to prove Lemma 4 by Han Bao [1, Lemma 3.1]. We here remark that in the error analysis of ours (and also of Goldstein), the argument of Schatz [7] plays an essential role, since the Helmholtz equation is indefinite. Analysis of the truncation error in the DtN finite element method is an important topic. For problems of the positive definite type, Yu [3] and Han Wu [13] have first derived error estimates including the truncation error for the exterior Laplace problem in After that, such error estimates for other problems are established in many papers, for example, in Han Wu [14] for the linear elastic problem, and in Givoli Patlashenko Keller [1] and Han Bao [1] for a certain class of the linear elliptic second order boundary value problem on exterior domains and semi-infinite strips (the error estimate in [1] is more sophisticated than that in [1]). For problems of other types, we mention Lenoir Tounsi [1] (the sea-keeping problem) and Koyama Tanimoto Ushijima [] (the eigenvalue problem of the linear water wave in a water region with a reentrant corner). The remainder of this paper is organized as follows. In Section, we formulate the reduced problem with the DtN boundary condition. In Section 3, we introduce some properties of the Hankel functions, which are employed for establishing the error estimates in Section 4.. The DtN formulation We first introduce a theorem concerning the well-posedness of problem (1). Theorem 1. For every compactly supported f L (Ω), problem (1) has a unique solution in Hloc (Ω), where H m loc (Ω) ={u u H m (B) for all bounded open set B Ω} (m N). Here L (Ω) denotes the usual space of complex-valued square integrable functions on Ω, and H m (B) denotes the usual complex Sobolev space on B (see, e.g., []). Proof. See [5,6].
3 D. Koyama / Journal of Computational and Applied Mathematics (7) To seek approximate solutions of problem (1), we introduce an artificial boundary Γ a ={x R d x =a}, where a is a positive number such that O supp f B a {x R d x <a}. Then problem (1) is reduced equivalently to the following problem on the bounded computational domain Ω a = Ω B a : Δu k u = f in Ω a, u = on γ, () u r = Su on Γ a, where S is the DtN operator corresponding to the outgoing radiation condition. For its definition, see [4,18]. Now we introduce the circular harmonics Y n defined by Y n (θ) = einθ π, where θ denotes the angular variable of an (r, θ) polar coordinate system, and the spherical harmonics Yn m Yn m (θ, ) = (n + 1) (n m )! 4π (n + m )! P n m (cos θ)e im, defined by where θ, denote the angular variables of an (r, θ, ) spherical coordinate system, and Pn m functions. We define the Sobolev space H s (Γ a ) (s>) by H s (Γ a ) ={φ L (Γ a ) φ s,γa < }, where s,γa is the norm of H s (Γ a ) defined by a (1 + n s ) φ n if d =, φ n= s,γ a = a n (1 + n s ) φ m n if d = 3, n= m= n where φ n and φ m n, respectively, are the Fourier coefficients of φ defined by π φ n = φ(θ)y n (θ) dθ and are the associated Legendre π π φ m n = d φ(θ, )Yn m (θ, ) sin θ dθ. (4) We here note that the DtN operator S is a bounded linear operator from H 1/ (Γ a ) into H 1/ (Γ a ) (see [4]), where H 1/ (Γ a ) is the set of all bounded semilinear forms on H 1/ (Γ a ). To formulate a weak form of problem (), we introduce the following sesquilinear forms: a(u, v) = ( u v k uv)dx + s(u, v) for u, v H 1 (Ω a ), Ω a (3) s(u, v) = Su, v H 1/ (Γ a ) H 1/ (Γ a ) n= = n n= m= n ka H n (1) (ka) H n (1) (ka) u n(a)v n (a) if d =, ka h(1) n (ka) h (1) n (ka) um n (a)vm n (a) if d = 3, (5)
4 4 D. Koyama / Journal of Computational and Applied Mathematics (7) 1 31 where H n (1) and h n (1) are the cylindrical and spherical Hankel functions of the first kind of order n, respectively, and u n (a) and u m n (a) denote the Fourier coefficients of u Γ a defined by (3) and (4), respectively. Then a weak form of () is written as follows: find u V such that a(u, v) = (f, v) (6) for all v V, where V ={v H 1 (Ω a ) v = onγ}, (u, v) = uv dx for u, v L (Ω a ). Ω a For every f L (Ω a ), problem (6) has a unique solution which is the restriction to Ω a of the solution of problem (1) (see [4,15,16]). 3. Some properties of the Hankel functions In this section, we state three lemmas concerning properties of the Hankel functions. These lemmas will be used to establish error estimates in the next section. Lemma. For each x>, there exists a positive constant C such that 1 H n (1) (x) 1 + n H n (1) C for all n Z, (7) (x) 1 h (1) n (x) 1 + n h n (1) C for all n N {}, (8) (x) where C depends on x, but is independent of n. Proof. For proofs of (7) and (8), see [4] and [16], respectively. Lemma 3. For all x>, we have } {H ν (1) (x) Re H ν (1) < for all ν R, (x) } {h (1) n (x) Re h n (1) < for all n N {}. (x) Proof. See [19]. Lemma 4. For any r 1 >r >, H ν (1) (r 1 )/H ν (1) (r ) is a decreasing function of ν on [, ), and further h (1) n (r 1 )/h (1) n (r ) is a decreasing sequence of n N {}. Proof. We first prove the former assertion. We write F(ν; r) = H ν (1) (r). From Nicholson s formula (see [9, p. 444]) F(ν; r) = 8 π K (r sinh t)cosh(νt)dt (ν R,r>), (9)
5 D. Koyama / Journal of Computational and Applied Mathematics (7) where K is the modified Bessel function of the second kind of order zero, we see that, for each r>, F( ;r) C (R) and F( ;r)>. Thus, it is sufficient to show that ( ) d F(ν; r1 ) < dν F(ν; r ) for all ν (, ). This inequality holds, if and only if df dν (ν; r 1)F (ν; r ) F(ν; r 1 ) df dν (ν; r )<. (1) Hence, let us prove (1) in the following. Differentiating (9) with ν leads to df 16 (ν; r) = dν π K (r sinh t)t sinh(νt)dt. (11) From (9) and (11), we have df dν (ν; r 1)F (ν; r ) F(ν; r 1 ) df dν (ν; r ) = 18 π 4 [K (r 1 sinh t 1 )K (r sinh t )t 1 sinh(νt 1 ) cosh(νt ) K (r sinh t 1 )K (r 1 sinh t )t 1 sinh(νt 1 ) cosh(νt )] dt 1 dt = 18 t1 π 4 dt 1 [K (r 1 sinh t 1 )K (r sinh t ) K (r 1 sinh t )K (r sinh t 1 )] [t 1 sinh(νt 1 ) cosh(νt ) t sinh(νt ) cosh(νt 1 )] dt. (1) Here, using Macdnold s formula (see [9, p. 439]) K (X)K (x) = 1 exp [ t X + x ] ( ) Xx dt K t t t and the fact that (X, x > ) (r 1 sinh t 1 + r sinh t ) (r 1 sinh t + r sinh t 1 ) = (r 1 r )(sinh t 1 sinh t )> for t 1 >t >, we can conclude that K (r 1 sinh t 1 )K (r sinh t ) K (r 1 sinh t )K (r sinh t 1 )< (13) for t 1 >t >. Further, we have t 1 sinh(νt 1 ) cosh(νt ) t sinh(νt ) cosh(νt 1 ) = 1 {(t 1 t ) sinh[ν(t 1 + t )]+(t 1 + t ) sinh[ν(t 1 t )]} > (14) for t 1 >t >. From (1) (14), we deduce (1). We next prove the latter assertion. Since h (1) n (r) = π r n+1/ (r), we have h n (1) (r 1 ) h n (1) (r ) = r r 1 n+1/ (r 1) n+1/ (r ). Hence, it follows from the above result that h (1) n (r 1 )/h (1) n (r ) is a decreasing sequence of n N {}.
6 6 D. Koyama / Journal of Computational and Applied Mathematics (7) The DtN finite element method and its error estimates We discretize problem (6) by using the finite element method to obtain approximate solutions to problem (6) (or (1)). We introduce a family {V h h (, h]} of finite dimensional subspaces of V, and assume that this family satisfies the following condition: there exist an integer p and a constant C> such that for all <h h and for every u V H p (Ω a )( p p), inf u v h 1,Ωa Ch p 1 u p v h,ω a, (15) V h where C is independent of h and u, and p,ωa is the standard norm of H p (Ω a ) defined by v p,ω = D α v dx (p N). Ω α p For examples of such a family, see [,36]. Now since the sesquilinear form s involves the infinite series, we have to truncate it in practice. So we practically solve the following problem: find u N h V h such that a N (u N h,v h) = (f, v h ) for all v h V h, (16) where, for N N, a N (u, v) = ( u v k uv)dx + s N (u, v) for u, v H 1 (Ω a ), Ω a ka H n (1) (ka) s N n <N H n (1) (ka) u n(a)v n (a) if d =, (u, v) = N 1 n ka h(1) n (ka) n= m= n h (1) n (ka) um n (a)vm n (a) if d = 3. Theorem 5. Let k be an arbitrary positive number and f an arbitrary function of L (Ω) with compact support. Assume that O supp f B a (a <a). Let u be the solution of problem (1). Assume that there exists an integer l such that u H l (Ω a ). Then there exist a γ > such that for every (h, N) (, h] N satisfying h + N 1 γ, problem (16) has a unique solution u N h, and moreover, if d =, then we have ( u u N h 1,Ω a C u u N h,ω a C(h + N 1 ) h m 1 u m,ωa + N s+1/ ( ) N (ka RN (u; s, a ), (17) ) N (ka RN (u; s, a ), (18) h m 1 u m,ωa + N s+1/ where,ωa is the usual norm of L (Ω a ), m = min{p, l}, s is an arbitrary real number 1, R N (u; s, a ) = a n N n s u n (a ), (19) and positive constants γ and C depend on k, a, and Ω a, but are independent of h, N, s, f, u, and u N h. If d = 3, then (17) and (18) hold by replacing N by h(1) N, and (19) by n R N (u; s, a ) = n s u m n (a ). a n N m= n
7 D. Koyama / Journal of Computational and Applied Mathematics (7) Before starting to prove Theorem 5, we introduce the following inequalities associated with the trace theorem: v m 1/,Γa C v m,ωa for all v H m (Ω a )(m= 1, ), () where C is a positive constant depending on Ω a, but independent of v, and the following sesquilinear form on H 1 (Ω a ): ka H n (1) (ka) r N n N H n (1) (ka) u n(a)v n (a) if d =, (u, v) = n ka h(1) n (ka) n N m= n h (1) n (ka) um n (a)vm n (a) if d = 3. Note here that we have s(u, v) = s N (u, v) + r N (u, v) for u, v H 1 (Ω a ). Proof. We prove only in the case when d =, because the proof of the case when d = 3 is exactly the same. We first assume that problem (16) has a solution u N h. We postpone proving the well-posedness of problem (16) until completion of the derivation of (17) and (18). Set eh N = u un h. Then we have a N (eh N,v h) + r N (u, v h ) = (1) for all v h V h. Note the following identical equation: e N h 1,Ω a = a N (e N h,en h ) + (k + 1) e N h,ω a s N (e N h,en h ). Taking the real part of this identity, we can get e N h 1,Ω a = Re{a N (e N h,en h )}+(k + 1) e N h,ω a Re{s N (e N h,en h )}. By virtue of Lemma 3, we have e N h 1,Ω a Re{a N (e N h,en h )}+(k + 1) e N h,ω a. () Step 1. We show that for an arbitrary ε >, there exists a positive constant C 3 (ε) such that a N (eh N,eN h ) ε en h 1,Ω a + C 3 (ε) hm u m,ω a + N s+1 )], (3) where s is an arbitrary number 1 and C 3(ε) depends on k, a, and Ω a, but is independent of h, N, s, u, and u N h.by (1), we have, for all v h V h, a N (e N h,en h ) = an (e N h,u v h) + r N (u, u N h v h) = a N (e N h,u v h) + r N (u, u v h ) r N (u, e N h ). Thus, by using the trigonometric inequality, the Schwarz inequality, Lemma, and the trace inequality (), we get a N (e N h,en h ) en h 1,Ω a u v h 1,Ωa + k e N h,ω a u v h,ωa + C(k,a) e N h 1/,Γ a u v h 1/,Γa + r N (u, u v h ) + r N (u, e N h ) C(k,Ω a ) eh N 1,Ω a u v h 1,Ωa + r N (u, u v h ) + r N (u, eh N ). (4) Let us estimate the second term on the right-hand side of (4). Since O supp f B a, the solution u can be analytically represented as follows u(r, θ) = n= n (kr) n (ka ) u n(a )Y n (θ) on R \B a.
8 8 D. Koyama / Journal of Computational and Applied Mathematics (7) 1 31 This implies u n (a) = H n (1) (ka) H n (1) (ka ) u n(a ) for all n Z. Moreover, we can see from the usual regularity argument that u Γa H s (Γ a ) for all s>. Thus, by the trigonometric inequality, Lemmas and 4, the Schwarz inequality, and the trace inequality (), we have, for every s 1, r N (u, u v h ) kah(1) n (ka) n N H n (1) (ka) u n(a) (u v h ) n (a) = n s+1/ ka H n (1) (ka) H n (1) (ka) n n N H n (1) (ka) H n (1) (ka ) n s u n (a ) n 1/ (u v h ) n (a) C(k,a,a )N s+1/ N (ka RN (u; s, a ) u v h 1/,Γa C(k,a, Ω a )N s+1/ N (ka RN (u; s, a ) u v h 1,Ωa, (5) where (u v h ) n (a) are the Fourier coefficients of u v h. In exactly the same way, we can estimate the third term on the right-hand side of (4) as follows: r N (u, eh N ) C(k,a, Ω a )N s+1/ N (ka RN (u; s, a ) eh N 1,Ω a. (6) Combining (4) (6) and (15) leads to [ a N (eh N,eN h ) C(k,a, Ω a ) h m 1 eh N 1,Ω a u m,ωa + h m 1 N s+1/ N (ka RN (u; s, a ) u m,ωa +N s+1/ ] N (ka en h 1,Ω a R N (u; s, a ). Applying the arithmetic geometric mean inequality to each term on the right-hand side of the above inequality, we obtain (3). Step. We show that there exists a positive constant C 4 such that eh N,Ω a C 4 [(h + N 1 ) eh N 1,Ω a + (hn s+1/ + N s 1/ N ) (ka) ] N (ka RN (u; s, a ), (7) where s is an arbitrary number 1 and C 4 depends on k, a, and Ω a, but is independent of h, N, s, u, and u N h. Suppose that w V satisfies a(v,w) = (v, e N h ) (8) for all v V. Then w is the incoming solution, that is, w is the restriction to Ω a of the solution of problem (1) where the outgoing radiation condition is replaced by the incoming radiation condition: ( ) u lim r + r(d 1)/ r + iku =
9 D. Koyama / Journal of Computational and Applied Mathematics (7) and f = eh N. Note here that the sesquilinear form s corresponding to the incoming radiation condition is represented by replacing H n (1) by H n () (the Hankel function of the second kind) in (5). Since Theorem 1 also holds for the incoming problem, we have w H (Ω a ) and the following a priori estimate: w,ωa C e N h,ω a, (9) where C is a positive constant independent of eh N. Taking v = en h in (8), we obtain eh N,Ω a = a(eh N,w)= an (eh N,w)+ rn (eh N,w). (3) Subtracting (1) from (3) gives e N h,ω a = a N (e N h,w v h) + r N (e N h,w) rn (u, v h ) = a N (eh N,w v h) + r N (eh N,w)+ rn (u, w v h ) r N (u, w). Employing the argument leading to (4), we can get e N h,ω a C(k,Ω a ) w v h 1,Ωa e N h 1,Ω a + r N (e N h,w) + rn (u, w v h ) + r N (u, w). (31) Employing an argument similar to the one used in (5), we can estimate the last three terms on the right-hand side of (31) as follows: r N (eh N,w) C(k,a)N 1 eh N 1/,Γ a w 3/,Γa C(k,Ω a )N 1 eh N 1,Ω a w,ωa, (3) r N (u, w v h ) C(k,a, Ω a )N s+1/ N (ka RN (u; s, a ) w v h 1,Ωa, (33) r N (u, w) C(k,a,a )N s 1/ N (ka RN (u; s, a ) w 3/,Γa C(k,a, Ω a )N s 1/ N (ka RN (u; s, a ) w,ωa. (34) Collecting (31) (34) yields {[ eh N,Ω a C(k,a, Ω a ) eh N 1,Ω a + N s+1/ ] N (ka RN (u; s, a ) w v h 1,Ωa [ + N 1 eh N 1,Ω a + N s 1/ ] } N (ka RN (u; s, a ) w,ωa. Using (15) and (9), we get {[ eh N,Ω a C(k,a, Ω a ) eh N 1,Ω a + N s+1/ ] N (ka RN (u; s, a ) h eh N,Ω a [ + N 1 eh N 1,Ω a + N s 1/ ] } N (ka RN (u; s, a ) e Nh,Ωa, and further dividing by eh N,Ω a, we obtain (7). Step 3. Let us collect the results above to get (17) and (18). Squaring both sides of (7) and using arithmetic-geometric mean inequality, we have eh N,Ω a C4 (h + N 1 ) eh N 1,Ω a + (hn s+1/ + N s 1/ ) )]. (35)
10 3 D. Koyama / Journal of Computational and Applied Mathematics (7) 1 31 Combining (), (3), and (35), we get eh N 1,Ω a ε eh N 1,Ω a + C 3 (ε) hm u m,ω a + N s+1 )] + C 5 (h + N 1 ) eh N 1,Ω a + (hn s+1/ + N s 1/ ) )], where C 5 = (k + 1)C4. This implies {1 ε C 5 (h + N 1 ) } eh N 1,Ω a C 6 (ε) h m u m,ω a + N s+1 )], where C 6 (ε) = C 3 (ε) + ( h + 1), and further, by taking ε = 1, { } 1 C 5(h + N 1 ) eh N 1,Ω a C 7 h m u m,ω a + N s+1 )], where C 7 = C 6 ( 1 ). For every {h, N} (, h] N satisfying 1 C 5(h + N 1 ) 4 1, which is equivalent to h + N 1 1 γ, 4C5 we have (17). Further, from (7) and (17), we can derive (18). Step 4. We finally show the well-posedness of problem (16). For this purpose, it is sufficient to prove uniqueness of the solution of problem (16) since V h is finite dimensional. Thus, assume now that u N h V h is a solution of problem (16) with f =. Since the solution u of problem (6) with f = is identically zero, it follows from (17) (or (18)) that u N h =. Therefore we can conclude that problem (16) is well-posed when h + N 1 γ. Acknowledgements The author would like to thank the referees for their comments leading to improvements of this paper. References [1] G. Bao, Finite element approximation of time harmonic waves in periodic structures, SIAM J. Numer. Anal. 3 (1995) [] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, [3] O.G. Ernst, A finite-element capacitance matrix method for exterior Helmholtz problems, Numer. Math. 75 () (1996) [4] K. Feng, Finite element method and natural boundary reduction, in: Proceedings of the International Congress of Mathematicians, Warszawa, 1983, pp [5] K. Feng, D.H. Yu, Canonical integral equations of elliptic boundary value problems and their numerical solutions, in: Proceedings of China France Symposium on the Finite Element Method (198, Beijing), Science Press, Beijing, 1983, pp [6] G.J. Fix, S.P. Marin, Variational methods for underwater acoustic problems, J. Comput. Phys. 8 () (1978) [7] D. Givoli, Numerical Methods for Problems in Infinite Domains, Studies in Applied Mechanics, vol. 33, Elsevier Scientific Publishing Co., Amsterdam, 199. [8] D. Givoli, Exact representations on artificial interfaces and applications in mechanics, Appl. Mech. Rev. 5 (1999) [9] D. Givoli, Book reviews, Appl. Mech. Rev. 56 (5) (3) B65. [1] D. Givoli, I. Patlashenko, J.B. Keller, High-order boundary conditions and finite elements for infinite domains, Comput. Methods Appl. Mech. Eng. 143 (1997)
11 D. Koyama / Journal of Computational and Applied Mathematics (7) [11] C.I. Goldstein, A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains, Math. Comp. 39 (198) [1] H. Han, W. Bao, Error estimates for the finite element approximation of problems in unbounded domains, SIAM J. Numer. Anal. 37 (4) () [13] H. Han, X. Wu, Approximation of infinite boundary condition and its application to finite element methods, J. Comput. Math. 3 (1985) [14] H. Han, X. Wu, The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its application, Math. Comp. 59 (199) [15] I. Harari, T.J.R. Hughes, Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Comput. Methods Appl. Mech. Eng. 97 (199) [16] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Applied Mathematical Sciences, vol. 13, Springer, New York, [17] Z.P. Jia, J.M. Wu, D.H. Yu, The coupling natural boundary-finite element method for solving 3-D exterior Helmholtz problem, Math. Numer. Sinica 3 (3) (1) (in Chinese). [18] J.B. Keller, D. Givoli, Exact nonreflecting boundary conditions, J. Comput. Phys. 8 (1989) [19] D. Koyama, A controllability method with an artificial boundary condition for the exterior Helmholtz problem, Japan J. Indust. Appl. Math. (1) (3) [] D. Koyama, K. Tanimoto, T. Ushijima, Finite element analysis for the eigenvalue problem of the linear water wave in a water region with a reentrant corner, Japan J. Indust. Appl. Math. 15 (3) (1998) [1] M. Lenoir, A. Tounsi, The localized finite element method and its application to the two-dimensional sea-keeping problem, SIAM J. Numer. Anal. 5 (4) (1988) [] J.L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. I, Springer, New York, 197. [3] R.C. MacCamy, S.P. Marin, A finite element method for exterior interface problems, Int. J. Math. Math. Sci. 3 () (198) [4] M. Masmoudi, Numerical solution for exterior problems, Numer. Math. 51 (1987) [5] R.S. Phillips, On the exterior problem for the reduced wave equation, in: Partial Differential Equations, Proceedings of the Symposium on Pure Mathematics, vol. XXIII, University of California, Berkeley, CA, 1971, American Mathematical Society, Providence, RI, 1973, pp [6] J. Sanchez-Hubert, E. Sanchez-Palencia, Vibration and Coupling of Continuous Systems, Asymptotic Methods, Springer, Berlin, New York, [7] A.H. Schatz, An observation concerning Ritz Galerkin methods with indefinite bilinear forms, Math. Comp. 8 (18) (1974) [8] T. Ushijima, An FEM-CSM combined method for planar exterior Laplace problems, Japan J. Indust. Appl. Math. 18 (1) [9] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge at the University Press, Cambridge, 19. [3] J.M. Wu, D.H. Yu, The natural integral equation of 3-D exterior Helmholtz problem and its numerical solution, Chinese J. Comp. Physics 16 (5) (1999) (in Chinese). [31] D.H. Yu, Coupling canonical boundary element method with FEM to solve harmonic problem over cracked domain, J. Comput. Math. 1 (3) (1983) 195. [3] D.H. Yu, Approximation of boundary conditions at infinity for a harmonic equation, J. Comput. Math. 3 (3) (1985) [33] D.H. Yu, Mathematical Theory of Natural Boundary Element Method, Science Press, Beijing, 1993 (in Chinese). [34] D.H. Yu, Natural Boundary Integral Method and its Applications, Mathematics and its Applications, vol. 539, Kluwer Academic Publishers, Science Press, Dordrecht, Beijing,. [35] D.H. Yu, Z.P. Jia, The non-overlapping DDM based on natural boundary reduction for -D exterior Helmholtz problem, Math. Numer. Sinica () () 7 4; D.H. Yu, Z.P. Jia, The non-overlapping DDM based on natural boundary reduction for -D exterior Helmholtz problem, Chinese J. Numer. Math. Appl. (3) () [36] M. Zlámal, Curved elements in the finite element method. I, SIAM J. Numer. Anal. 1 (1973) 9 4.
A Parallel Schwarz Method for Multiple Scattering Problems
A Parallel Schwarz Method for Multiple Scattering Problems Daisuke Koyama The University of Electro-Communications, Chofu, Japan, koyama@imuecacjp 1 Introduction Multiple scattering of waves is one of
More informationThe Coupling Method with the Natural Boundary Reduction on an Ellipse for Exterior Anisotropic Problems 1
Copyright 211 Tech Science Press CMES, vol.72, no.2, pp.13-113, 211 The Coupling Method with the Natural Boundary Reduction on an Ellipse for Exterior Anisotropic Problems 1 Quan Zheng 2, Jing Wang 2 and
More informationError analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis
Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis George C. Hsiao a, Nilima Nigam b, Joseph E. Pasciak c, Liwei Xu d, a Department of Mathematical Sciences, University
More informationON THE COUPLING OF BEM AND FEM FOR EXTERIOR PROBLEMS FOR THE HELMHOLTZ EQUATION
MATHEMATICS OF COMPUTATION Volume 68, Number 227, Pages 945 953 S 0025-5718(99)01064-9 Article electronically published on February 15, 1999 ON THE COUPLING OF BEM AND FEM FOR EXTERIOR PROBLEMS FOR THE
More informationA new method for the solution of scattering problems
A new method for the solution of scattering problems Thorsten Hohage, Frank Schmidt and Lin Zschiedrich Konrad-Zuse-Zentrum Berlin, hohage@zibde * after February 22: University of Göttingen Abstract We
More informationError bounds for the finite-element approximation of the exterior Stokes equations in two dimensions
IMA Journal of Numerical Analysis 3) 3, 5 47 Error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions WEIZHU BAO Department of Computational Science, National
More informationFinite Element Analysis of Acoustic Scattering
Frank Ihlenburg Finite Element Analysis of Acoustic Scattering With 88 Illustrations Springer Contents Preface vii 1 The Governing Equations of Time-Harmonic Wave Propagation, 1 1.1 Acoustic Waves 1 1.1.1
More informationJournal of Computational and Applied Mathematics. Three-dimensional approximate local DtN boundary conditions for prolate spheroid boundaries
Journal of Computational and Applied Mathematics 34 (010) 1810 1816 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
More informationAn Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition
Noname manuscript No. will be inserted by the editor An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition Xue Jiang Peijun Li Junliang Lv Weiying Zheng eceived:
More informationMathematical Foundations for the Boundary- Field Equation Methods in Acoustic and Electromagnetic Scattering
Mathematical Foundations for the Boundary- Field Equation Methods in Acoustic and Electromagnetic Scattering George C. Hsiao Abstract The essence of the boundary-field equation method is the reduction
More informationACMAC s PrePrint Repository
ACMAC s PrePrint Repository Helmholtz equation with artificial boundary conditions in a twodimensional waveguide D. A. Mitsoudis and Ch. Makridakis and M. Plexousakis Original Citation: Mitsoudis, D. A.
More informationShape design problem of waveguide by controlling resonance
DD3 Domain Decomposition 3, June 6-10, 015 at ICC-Jeju, Jeju Island, Korea Shape design problem of waveguide by controlling resonance KAKO, Takashi Professor Emeritus & Industry-academia-government collaboration
More informationThe Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator
The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator Martin J. Gander and Achim Schädle Mathematics Section, University of Geneva, CH-, Geneva, Switzerland, Martin.gander@unige.ch
More informationNatural Boundary Integral Method and Its Applications
Natural Boundary Integral Method and Its Applications By De-hao Yu State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing
More informationA coupled BEM and FEM for the interior transmission problem
A coupled BEM and FEM for the interior transmission problem George C. Hsiao, Liwei Xu, Fengshan Liu, Jiguang Sun Abstract The interior transmission problem (ITP) is a boundary value problem arising in
More informationOn spherical-wave scattering by a spherical scatterer and related near-field inverse problems
IMA Journal of Applied Mathematics (2001) 66, 539 549 On spherical-wave scattering by a spherical scatterer and related near-field inverse problems C. ATHANASIADIS Department of Mathematics, University
More informationAdditive Schwarz method for scattering problems using the PML method at interfaces
Additive Schwarz method for scattering problems using the PML method at interfaces Achim Schädle 1 and Lin Zschiedrich 2 1 Zuse Institute, Takustr. 7, 14195 Berlin, Germany schaedle@zib.de 2 Zuse Institute,
More informationAnalysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions
Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions Seungil Kim Department of Mathematics and Research Institute for Basic Sciences,
More informationMANY scientific and engineering problems can be
A Domain Decomposition Metod using Elliptical Arc Artificial Boundary for Exterior Problems Yajun Cen, and Qikui Du Abstract In tis paper, a Diriclet-Neumann alternating metod using elliptical arc artificial
More informationA mixed nite volume element method based on rectangular mesh for biharmonic equations
Journal of Computational and Applied Mathematics 7 () 7 3 www.elsevier.com/locate/cam A mixed nite volume element method based on rectangular mesh for biharmonic equations Tongke Wang College of Mathematical
More informationANALYSIS OF A FINITE ELEMENT PML APPROXIMATION FOR THE THREE DIMENSIONAL TIME-HARMONIC MAXWELL PROBLEM
MATHEMATICS OF COMPUTATION Volume 77, Number 261, January 2008, Pages 1 10 S 0025-5718(07)02037-6 Article electronically published on September 18, 2007 ANALYSIS OF A FINITE ELEMENT PML APPROXIMATION FOR
More informationNEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION IN A SHALLOW OCEAN WITH FLUID-LIKE SEABED
Georgian Mathematical Journal Volume 4 2007, Number, 09 22 NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION IN A SHALLOW OCEAN WITH FLUID-LIKE SEAED ROERT GILERT AND MIAO-JUNG OU Abstract. In
More informationA Finite Element Method Using Singular Functions for Poisson Equations: Mixed Boundary Conditions
A Finite Element Method Using Singular Functions for Poisson Equations: Mixed Boundary Conditions Zhiqiang Cai Seokchan Kim Sangdong Kim Sooryun Kong Abstract In [7], we proposed a new finite element method
More information44 CHAPTER 3. CAVITY SCATTERING
44 CHAPTER 3. CAVITY SCATTERING For the TE polarization case, the incident wave has the electric field parallel to the x 3 -axis, which is also the infinite axis of the aperture. Since both the incident
More informationRiesz bases of Floquet modes in semi-infinite periodic waveguides and implications
Riesz bases of Floquet modes in semi-infinite periodic waveguides and implications Thorsten Hohage joint work with Sofiane Soussi Institut für Numerische und Angewandte Mathematik Georg-August-Universität
More informationAbstract. 1. Introduction
Journal of Computational Mathematics Vol.28, No.2, 2010, 273 288. http://www.global-sci.org/jcm doi:10.4208/jcm.2009.10-m2870 UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS
More informationFactorization method in inverse
Title: Name: Affil./Addr.: Factorization method in inverse scattering Armin Lechleiter University of Bremen Zentrum für Technomathematik Bibliothekstr. 1 28359 Bremen Germany Phone: +49 (421) 218-63891
More informationA Posteriori Error Estimation for Highly Indefinite Helmholtz Problems
Computational Methods in Applied Mathematics Vol. 13 013), No. 3, pp. 333 347 c 013 Institute of Mathematics, NAS of Belarus Doi: 10.1515/cmam-013-0008 A Posteriori Error Estimation for Highly Indefinite
More informationOn an Approximation Result for Piecewise Polynomial Functions. O. Karakashian
BULLETIN OF THE GREE MATHEMATICAL SOCIETY Volume 57, 010 (1 7) On an Approximation Result for Piecewise Polynomial Functions O. arakashian Abstract We provide a new approach for proving approximation results
More informationStabilization and Controllability for the Transmission Wave Equation
1900 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 12, DECEMBER 2001 Stabilization Controllability for the Transmission Wave Equation Weijiu Liu Abstract In this paper, we address the problem of
More informationNATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY
NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY Hong-ying Huang De-hao Yu State Key Laboratory of Scientific Engineering Computing,
More information1. How do error estimators for the Galerkin FEM depend on &, ft? 2. Can the classical Galerkin approach be improved towards a linear rule?
Reliability of finite element methods for the numerical computation of waves F. Ihlenburg*, I. Babuska*, S. Sauted "Institute for Physical Science and Technology, University of Maryland at College Park,
More informationStrong Stabilization of the System of Linear Elasticity by a Dirichlet Boundary Feedback
To appear in IMA J. Appl. Math. Strong Stabilization of the System of Linear Elasticity by a Dirichlet Boundary Feedback Wei-Jiu Liu and Miroslav Krstić Department of AMES University of California at San
More informationUniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers
INSTITUTE OF PHYSICS PUBLISHING Inverse Problems 22 (2006) 515 524 INVERSE PROBLEMS doi:10.1088/0266-5611/22/2/008 Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and
More informationMultiscale methods for time-harmonic acoustic and elastic wave propagation
Multiscale methods for time-harmonic acoustic and elastic wave propagation Dietmar Gallistl (joint work with D. Brown and D. Peterseim) Institut für Angewandte und Numerische Mathematik Karlsruher Institut
More informationMaximum norm estimates for energy-corrected finite element method
Maximum norm estimates for energy-corrected finite element method Piotr Swierczynski 1 and Barbara Wohlmuth 1 Technical University of Munich, Institute for Numerical Mathematics, piotr.swierczynski@ma.tum.de,
More informationCONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION
Journal of Computational Acoustics, Vol. 8, No. 1 (2) 139 156 c IMACS CONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION MURTHY N. GUDDATI Department of Civil Engineering, North Carolina
More informationarxiv: v1 [math.na] 29 Feb 2016
EFFECTIVE IMPLEMENTATION OF THE WEAK GALERKIN FINITE ELEMENT METHODS FOR THE BIHARMONIC EQUATION LIN MU, JUNPING WANG, AND XIU YE Abstract. arxiv:1602.08817v1 [math.na] 29 Feb 2016 The weak Galerkin (WG)
More informationA Least-Squares Finite Element Approximation for the Compressible Stokes Equations
A Least-Squares Finite Element Approximation for the Compressible Stokes Equations Zhiqiang Cai, 1 Xiu Ye 1 Department of Mathematics, Purdue University, 1395 Mathematical Science Building, West Lafayette,
More informationError analysis and preconditioning for an enhanced DtN-FE algorithm for exterior scattering problems
Journal of Computational and Applied Mathematics 24 (27) 493 54 www.elsevier.com/locate/cam Error analysis and preconditioning for an enhanced DtN-FE algorithm for exterior scattering problems Leonid Chindelevitch
More informationStrong stabilization of the system of linear elasticity by a Dirichlet boundary feedback
IMA Journal of Applied Mathematics (2000) 65, 109 121 Strong stabilization of the system of linear elasticity by a Dirichlet boundary feedback WEI-JIU LIU AND MIROSLAV KRSTIĆ Department of AMES, University
More informationGlobal Solutions for a Nonlinear Wave Equation with the p-laplacian Operator
Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática
More informationMixed exterior Laplace s problem
Mixed exterior Laplace s problem Chérif Amrouche, Florian Bonzom Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des Pays de l Adour, IPRA, Avenue de l Université, 64000 Pau
More informationInverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing
Inverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing Isaac Harris Texas A & M University College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work with: F. Cakoni, H.
More informationTrefftz type method for 2D problems of electromagnetic scattering from inhomogeneous bodies.
Trefftz type method for 2D problems of electromagnetic scattering from inhomogeneous bodies. S. Yu. Reutsiy Magnetohydrodynamic Laboratory, P. O. Box 136, Mosovsi av.,199, 61037, Kharov, Uraine. e mail:
More informationarxiv: v1 [math.na] 15 Nov 2017
Noname manuscript No. (will be inserted by the editor) An HDG method with orthogonal projections in facet integrals Issei Oikawa arxiv:1711.05396v1 [math.na] 15 Nov 2017 Received: date / Accepted: date
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationTwo dimensional exterior mixed problem for semilinear damped wave equations
J. Math. Anal. Appl. 31 (25) 366 377 www.elsevier.com/locate/jmaa Two dimensional exterior mixed problem for semilinear damped wave equations Ryo Ikehata 1 Department of Mathematics, Graduate School of
More informationTechnische Universität Graz
Technische Universität Graz Stability of the Laplace single layer boundary integral operator in Sobolev spaces O. Steinbach Berichte aus dem Institut für Numerische Mathematik Bericht 2016/2 Technische
More informationVariational Formulations
Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that
More informationTrefftz-discontinuous Galerkin methods for time-harmonic wave problems
Trefftz-discontinuous Galerkin methods for time-harmonic wave problems Ilaria Perugia Dipartimento di Matematica - Università di Pavia (Italy) http://www-dimat.unipv.it/perugia Joint work with Ralf Hiptmair,
More informationAn ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains
c de Gruyter 2009 J. Inv. Ill-Posed Problems 17 (2009), 703 711 DOI 10.1515 / JIIP.2009.041 An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains W. Arendt and T. Regińska
More informationPARTITION OF UNITY FOR THE STOKES PROBLEM ON NONMATCHING GRIDS
PARTITION OF UNITY FOR THE STOES PROBLEM ON NONMATCHING GRIDS CONSTANTIN BACUTA AND JINCHAO XU Abstract. We consider the Stokes Problem on a plane polygonal domain Ω R 2. We propose a finite element method
More informationThe mathematics of scattering by unbounded, rough, inhomogeneous layers
The mathematics of scattering by unbounded, rough, inhomogeneous layers Simon N. Chandler-Wilde a Peter Monk b Martin Thomas a a Department of Mathematics, University of Reading, Whiteknights, PO Box 220
More informationA Mortar Element Method for Coupling Natural Boundary Element Method and Finite Element Method for Unbounded Domain Problem
Contemporary Mathematics A Mortar Element Method for Coupling Natural Boundary Element Method and Finite Element Method for Unbounded Domain Problem Sheng Zhang and Dehao Yu Abstract. In this paper, a
More informationBoundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis
Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis T. Tran Q. T. Le Gia I. H. Sloan E. P. Stephan Abstract Radial basis functions are used to define approximate solutions
More informationInternal Stabilizability of Some Diffusive Models
Journal of Mathematical Analysis and Applications 265, 91 12 (22) doi:1.16/jmaa.21.7694, available online at http://www.idealibrary.com on Internal Stabilizability of Some Diffusive Models Bedr Eddine
More informationCoercivity of high-frequency scattering problems
Coercivity of high-frequency scattering problems Valery Smyshlyaev Department of Mathematics, University College London Joint work with: Euan Spence (Bath), Ilia Kamotski (UCL); Comm Pure Appl Math 2015.
More informationOutgoing wave conditions in photonic crystals and transmission properties at interfaces
Outgoing wave conditions in photonic crystals and transmission properties at interfaces A. Lamacz, B. Schweizer December 19, 216 Abstract We analyze the propagation of waves in unbounded photonic crystals.
More informationUniqueness in determining refractive indices by formally determined far-field data
Applicable Analysis, 2015 Vol. 94, No. 6, 1259 1269, http://dx.doi.org/10.1080/00036811.2014.924215 Uniqueness in determining refractive indices by formally determined far-field data Guanghui Hu a, Jingzhi
More informationWEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER PARABOLIC EQUATIONS
INERNAIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 13, Number 4, Pages 525 544 c 216 Institute for Scientific Computing and Information WEAK GALERKIN FINIE ELEMEN MEHOD FOR SECOND ORDER PARABOLIC
More informationarxiv: v1 [math.na] 27 Jan 2016
Virtual Element Method for fourth order problems: L 2 estimates Claudia Chinosi a, L. Donatella Marini b arxiv:1601.07484v1 [math.na] 27 Jan 2016 a Dipartimento di Scienze e Innovazione Tecnologica, Università
More informationINFINITE ELEMENT METHODS FOR HELMHOLTZ EQUATION PROBLEMS ON UNBOUNDED DOMAINS
INFINITE ELEMENT METHODS FOR HELMHOLTZ EQUATION PROBLEMS ON UNBOUNDED DOMAINS Michael Newman Department of Aerospace Engineering Texas A&M University 34 TAMU 744D H.R. Bright Building College Station,
More informationMIXED BOUNDARY-VALUE PROBLEMS FOR QUANTUM HYDRODYNAMIC MODELS WITH SEMICONDUCTORS IN THERMAL EQUILIBRIUM
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 123, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) MIXED
More informationWavelet moment method for Cauchy problem for the Helmholtz equation
Wavelet moment method for Cauchy problem for the Helmholtz equation Preliminary report Teresa Regińska, Andrzej Wakulicz December 9, 2005 - Institute of Mathematics, Polish Academy of Sciences, e-mail:
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationSECOND ORDER TIME DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS
Proceedings of ALGORITMY 2009 pp. 1 10 SECOND ORDER TIME DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS MILOSLAV VLASÁK Abstract. We deal with a numerical solution of a scalar
More informationOn the Spectrum of Volume Integral Operators in Acoustic Scattering
11 On the Spectrum of Volume Integral Operators in Acoustic Scattering M. Costabel IRMAR, Université de Rennes 1, France; martin.costabel@univ-rennes1.fr 11.1 Volume Integral Equations in Acoustic Scattering
More informationA far-field based T-matrix method for three dimensional acoustic scattering
ANZIAM J. 50 (CTAC2008) pp.c121 C136, 2008 C121 A far-field based T-matrix method for three dimensional acoustic scattering M. Ganesh 1 S. C. Hawkins 2 (Received 14 August 2008; revised 4 October 2008)
More informationA Priori Error Analysis for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems
A Priori Error Analysis for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems D. Meidner and B. Vexler Abstract In this article we discuss a priori error estimates for Galerkin
More informationApplied Numerical Mathematics
Applied Numerical Mathematics 131 (2018) 1 15 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum The spectral-galerkin approximation of nonlinear eigenvalue
More informationDomain decomposition schemes with high-order accuracy and unconditional stability
Domain decomposition schemes with high-order accuracy and unconditional stability Wenrui Hao Shaohong Zhu March 7, 0 Abstract Parallel finite difference schemes with high-order accuracy and unconditional
More informationPreconditioned space-time boundary element methods for the heat equation
W I S S E N T E C H N I K L E I D E N S C H A F T Preconditioned space-time boundary element methods for the heat equation S. Dohr and O. Steinbach Institut für Numerische Mathematik Space-Time Methods
More informationAge-time continuous Galerkin methods for a model of population dynamics
Journal of Computational and Applied Mathematics 223 (29) 659 671 www.elsevier.com/locate/cam Age-time continuous Galerkin methods for a model of population dynamics Mi-Young Kim, Tsendauysh Selenge Department
More informationOptimal shape and position of the support for the internal exact control of a string
Optimal shape and position of the support for the internal exact control of a string Francisco Periago Abstract In this paper, we consider the problem of optimizing the shape and position of the support
More informationThe Factorization Method for Inverse Scattering Problems Part I
The Factorization Method for Inverse Scattering Problems Part I Andreas Kirsch Madrid 2011 Department of Mathematics KIT University of the State of Baden-Württemberg and National Large-scale Research Center
More informationON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON
NANOSYSTES: PHYSICS, CHEISTRY, ATHEATICS, 15, 6 (1, P. 46 56 ON THE ROBIN EIGENVALUES OF THE LAPLACIAN IN THE EXTERIOR OF A CONVEX POLYGON Konstantin Pankrashkin Laboratoire de mathématiques, Université
More informationTWO-SCALE CONVERGENCE ON PERIODIC SURFACES AND APPLICATIONS
TWO-SCALE CONVERGENCE ON PERIODIC SURFACES AND APPLICATIONS Grégoire ALLAIRE Commissariat à l Energie Atomique DRN/DMT/SERMA, C.E. Saclay 91191 Gif sur Yvette, France Laboratoire d Analyse Numérique, Université
More informationASYMPTOTICALLY EXACT A POSTERIORI ESTIMATORS FOR THE POINTWISE GRADIENT ERROR ON EACH ELEMENT IN IRREGULAR MESHES. PART II: THE PIECEWISE LINEAR CASE
MATEMATICS OF COMPUTATION Volume 73, Number 246, Pages 517 523 S 0025-5718(0301570-9 Article electronically published on June 17, 2003 ASYMPTOTICALLY EXACT A POSTERIORI ESTIMATORS FOR TE POINTWISE GRADIENT
More informationSpectrum and Exact Controllability of a Hybrid System of Elasticity.
Spectrum and Exact Controllability of a Hybrid System of Elasticity. D. Mercier, January 16, 28 Abstract We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped
More informationSurvey of Inverse Problems For Hyperbolic PDEs
Survey of Inverse Problems For Hyperbolic PDEs Rakesh Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA Email: rakesh@math.udel.edu January 14, 2011 1 Problem Formulation
More informationTechnische Universität Graz
Technische Universität Graz A note on the stable coupling of finite and boundary elements O. Steinbach Berichte aus dem Institut für Numerische Mathematik Bericht 2009/4 Technische Universität Graz A
More informationSUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS
SUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS A. RÖSCH AND R. SIMON Abstract. An optimal control problem for an elliptic equation
More informationA REMARK ON MINIMAL NODAL SOLUTIONS OF AN ELLIPTIC PROBLEM IN A BALL. Olaf Torné. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 24, 2004, 199 207 A REMARK ON MINIMAL NODAL SOLUTIONS OF AN ELLIPTIC PROBLEM IN A BALL Olaf Torné (Submitted by Michel
More informationMULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH
MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH MARCELO F. FURTADO AND HENRIQUE R. ZANATA Abstract. We prove the existence of infinitely many solutions for the Kirchhoff
More informationBessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy
More informationMULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2015 (2015), o. 274, pp. 1 9. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIOS
More informationPDEs, part 1: Introduction and elliptic PDEs
PDEs, part 1: Introduction and elliptic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Partial di erential equations The solution depends on several variables,
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationTrefftz-Discontinuous Galerkin Methods for Acoustic Scattering - Recent Advances
Trefftz-Discontinuous Galerkin Methods for Acoustic Scattering - Recent Advances Ilaria Perugia Dipartimento di Matematica Università di Pavia (Italy) In collaboration with Ralf Hiptmair, Christoph Schwab,
More informationStress Analysis of Elastic Roof with Boundary Element Formulations
Copyright 211 Tech Science Press CMC, vol.26, no.3, pp.187-2, 211 Stress Analysis of Elastic Roof with Boundary Element Formulations Dan Ma 1,2 and Xianbiao Mao 1 Abstract: Roof is one of the most important
More informationOn the remainder term of Gauss Radau quadratures for analytic functions
Journal of Computational and Applied Mathematics 218 2008) 281 289 www.elsevier.com/locate/cam On the remainder term of Gauss Radau quadratures for analytic functions Gradimir V. Milovanović a,1, Miodrag
More informationA Simple Compact Fourth-Order Poisson Solver on Polar Geometry
Journal of Computational Physics 182, 337 345 (2002) doi:10.1006/jcph.2002.7172 A Simple Compact Fourth-Order Poisson Solver on Polar Geometry Ming-Chih Lai Department of Applied Mathematics, National
More informationCOMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute
More informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More informationStability of an abstract wave equation with delay and a Kelvin Voigt damping
Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability
More informationBulletin of the. Iranian Mathematical Society
ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Bulletin of the Iranian Mathematical Society Vol. 42 (2016), No. 1, pp. 129 141. Title: On nonlocal elliptic system of p-kirchhoff-type in Author(s): L.
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationON THE EXISTENCE OF TRANSMISSION EIGENVALUES. Andreas Kirsch1
Manuscript submitted to AIMS Journals Volume 3, Number 2, May 29 Website: http://aimsciences.org pp. 1 XX ON THE EXISTENCE OF TRANSMISSION EIGENVALUES Andreas Kirsch1 University of Karlsruhe epartment
More information