MANY scientific and engineering problems can be

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1 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 boundary is designed to solve exterior Poisson problem wit a concave angle. It is sown tat te metod is equivalent to a preconditioned Ricardson iteration. Te convergence of tis metod and its discretization are studied. Finally, some numerical examples are given to sow te effectiveness of tis metod. Index Terms Diriclet-Neumann alternating metod, elliptical arc artificial boundary, exterior problem. Ω I. INTRODUCTION MANY scientific and engineering problems can be modeled by exterior boundary value problems of partial differential equations wic are required to be solved in unbounded domains. In te last tree decades, some metods for solving problems over unbounded domains ave been developed. One of te commonly used tecniques is te metod of artificial boundary conditions [1]-[9]. Te metod may be summarized as follows: (i) Introduce an artificial boundary µ, wic divides te original unbounded domain into two non-overlapping subdomains: a bounded computational domain Ω i and infinite residual domain Ω e. (ii) By analyzing te problem in Ω e, obtain a relation on µ (exact or approximate) involving te unknown function u and its derivatives. (iii) Using te relation as a boundary condition on µ, to obtain a well-posed problem in Ω i. (iv) Solve te problem in Ω i be te standard finite element metods or some oter numerical metods.te relation obtained in Step (ii) and used as a boundary condition in Step (iii) is called an artificial boundary condition. Based on artificial boundary conditions, te overlapping and non-overlapping domain decomposition metods can be viewed as effective ways to solve problems in unbounded domains. Tese tecniques ave been used to solve many linear or nonlinear problems [1]-[17]. Recently, te autors used a new elliptical arc artificial boundary to solve Poisson problems and anisotropic problems [18]-[19], and construct an iteration metod wic is equivalent to a Scwarz alternating metod to solve Poisson problems in concave angle domains []. In tis paper, we design a Diriclet-Neumann alternating metod based on an elliptical arc artificial boundary to solve exterior Poisson problem wit a concave angle. Ω be an exterior concave angle domain wit angle, and < π. Te boundary of domain Ω is decomposed Manuscript received Marc, 17; revised June 3, 17. Tis work was supported by te National Natural Science Foundation of Cina (Grant No ). Yajun Cen is wit te Scool of Matematical Sciences, Nanjing Normal University, Nanjing 13 and te Department of Matematics, Sangai Maritime University, Sangai 136, Cina ( cenyajun@smtu.edu.cn). Qikui Du is wit te Scool of Matematical Sciences, Nanjing Normal University, Nanjing 13, Cina. Fig. 1. Te Illustration of Domain Ω into tree disjoint parts:, and (see Fig. 1), i.e. Ω =, = Ø, = Ø, = Ø. Te boundary is a simple smoot curve part, and are two alf lines. We consider te Poisson problem in two cases: u = f, in Ω, u =, on, u (1) = g, on, u is vanis at infinity, and u = f, in Ω, u =, on, () u =, on, u is bounded at infinity, u is te unknown function, f L (Ω) and g, L () are given functions, supp(f) is compact. Te rest of te paper is organized as follows. In Section, we introduce an elliptical arc artificial boundary wic divide te original domain Ω into two non-overlapping subdomains, ten we construct a Diriclet-Neumann alternating metod. In Section 3, we give te weak form and discretization. In Section 4, we analyze te convergence of te metod. Finally, in Section 5 we present some numerical results to sow its accuracy and te effectiveness of our metod. II. DIRICHLET-NEUMANN ALTERNATING METHOD Draw a elliptical arc 1 = {(µ, φ) µ = µ 1, < φ < }, wic enclose suc tat dist(, 1 ) >. Ten Ω is divided into two non-overlapping subdomains and Ω (see Fig. ). be te bounded domain among,, and 1, and Ω be te unbounded domain outside 1, and. Ten te original problems (1) is decomposed into two subproblems in domains and Ω wit Ω = Ø, = 1 1 1, Ω = 1, i = Ω i, i = Ω i, i = 1,. (Advance online publication: 17 November 17)

2 In te first case, we proposed te Diriclet-Neumann alternating metod as follows. Step. Pick an initial value λ H 1 ( 1 ), and put k =. Step 1. Solve a Diriclet problem in Ω u k = f, in Ω, u k =, on, u k = λ k (3), on 1, u k is vanis at infinity. Ω 1 Step. Solve a mixed problem in u k 1 = f, in, u k 1 =, on 1 1, u k 1 = g, on, u k 1 = uk, on 1. Step 3. Update te boundary value on 1 by (4) λ k+1 = θ k u k 1 + (1 θ k )λ k. (5) Step 4. Set k = k + 1, ten goto Step 1. u k 1 and u k are te kt approximate solutions in and Ω, respectively. θ k denotes te kt relaxation factor and λ is an arbitrary function in H 1 ( 1 ). Note tat, on interface 1, only te value of te normal derivative of te solution of (3) is needed in solving (4). So it is unnecessary to solve (3). Actually, we can obtain uk directly from λk by making use of te following artificial boundary condition [18]: u k = Kλk, (6) Kλ k = π n λ k (µ 1, ϕ) sin nπφ nπϕ sin J dϕ. (7) For te second case, we can also construct te Diriclet- Neumann alternating metod. In te following sections, we just consider te discretization and convergence of problem (1), we can obtain corresponding result of problem () in te same way. III. THE WEAK FORM AND DISCRETIZATION V ( ) = {v v H 1 ( ), v 1 1 = }, ten problem (1) is equivalent to te variational problem: Find u V ( ), suc tat b(u, v) = a(u, v) + b(u, v) = f(v), v V ( ), (8) nπ a(u, v) = u vdx, (9) u v ϕ φ f(v) = fvdx + cos nπϕ nπφ cos dϕdφ, (1) gvds. (11) Fig.. Te Illustration of Domain and Ω S ( ) V ( ) denote te liner finite element space of V ( ). Ten te approximate variational problem of (8) can be written as: Find u S ( ), suc tat a(u, v ) + b(u, v ) = f(v ), v S ( ). (1) From te problem (1), we can get algebraic equations as follows A 11 + B A 1i A 1 A i1 A ii A i U 1 U i =, (13) A 1 A i A U F U 1 is a vector wose components are function values at nodes on 1, U i is a vector wose components are function values at interior nodes in and U is a vector wose components are function values at nodes on. Te matrix A = A 11 A 1i A 1 A i1 A ii A i A 1 A i A is te stiffness matrix obtained from finite element in wile B is gotten from te artificial boundary condition on 1. (13) can also be rewritten as follows A 11 A 1i A 1 A i1 A ii A i A 1 A i A U 1 U i U = Ten, we ave te iterative metod A 11 A 1i A 1 U1 k A i1 A ii A i U k i = A 1 A i A U k wit BU 1 F BΛk F. (14), (15) Λ k+1 = θ k U k 1 + (1 θ k )Λ k, k =, 1,,... (16) IV. ANALYSIS OF CONVERGENCE It is difficult to analyze te convergence of te above alternating metod in te general domain. However, te analysis is possible for some special curve. Terefore, we only consider te case te boundaries and 1 bot are elliptical arcs, i.e., = {(µ, φ) µ = µ, < φ < }, 1 = {(µ, φ) µ = µ 1, < φ < }, and µ 1 > µ. We first consider te convergence of te metod in continuous case. Teorem 1. If < θ k < 1, ten te Diriclet-Neumann alternating metod (3)-(5) is convergent. Proof. e k = λ λ k = b n sin nπφ, on 1, (Advance online publication: 17 November 17)

3 we ave e k 1 = K(ek ) = π By te separation of variables, we ave Hence Ten, we ave e k 1 = H n (µ) = K(e k 1) = π e k+1 1 = K(λ λ k+1 ) b n H n (µ) sin nπφ, e nπ (µ µ) e nπ (µ µ) e nπ (µ1 µ) e nπ = K(θ k u k 1 + (1 θ k )λ k λ) = π If we let ten and E k+1 = π nb n sin nπφ. (17) (µ µ1). nb n H n (µ) sin nπφ. nb n (θ k H n (µ 1 ) 1 + θ k ) sin nπφ. E k = ek 1 1,1, E k = π (1 + n ) 1 n b J n, (1 + n ) 1 n b n(θ k H n (µ 1 ) 1 + θ k ) = (1 θ k ) E k + π (1 + n ) 1 n b n θ k H n (µ 1 )(θ k H n (µ 1 ) + θ k ). δ = inf n Z + + H n (µ 1 ). A computation sows tat δ = 3. If we let θ k, k =, 1,,..., satisfy < θ k δ, ten By te trace teorem, we ave (18) E k+1 < (1 θ k ) E k. (19) e k 1 1, CE k, k +. Tis means tat te Diriclet-Neumann alternating metod is convergent if < θ k δ. We also ave E k+1 = π (1 + n ) 1 n b n(θ k 1 θ k G n (µ 1 )) = (1 θ k ) E k + π (1 + n ) 1 n b n θ k G n (µ 1 )(θ k G n (µ 1 ) θ k + 1), G n (µ 1 ) = 1 H n(µ 1 ). σ = sup n Z + 1 G n (µ 1 ). It is easy to get σ = 3. Similar to te above analysis, if we take θ k, k =, 1,,..., satisfy σ θ k < 1, te Diriclet-Neumann alternating metod is also convergent. Terefore, for < θ k < 1, te Diriclet-Neumann alternating metod is convergent. In te following, we consider te convergence of te discretization form. Teorem. Te discrete Diriclet-Neumann alternating metod (15) and (16) are equivalent to te following preconditioned Ricardson iteration: (Λk+1 Λ k ) = θ k (F 1 S Λ k ), () = A 11 A 1i (A ii A i (A ) 1 A i ) 1 A i1, (1) S = + B, () F 1 = A 1i (A ii A i (A ) 1 A i ) 1 A i (A ) 1 F. (3) Proof. From (13), we ave namely, (A 11 A 1i (A ii A i (A ) 1 A i ) 1 A i1 + B)U 1 = A 1i (A ii A i (A ) 1 A i ) 1 A i (A ) 1 F, S U 1 = F 1. From (14) and (15), we obtain U1 k U 1 A Ui k U i = B(Λk U 1 ). U k U So Terefore (U k 1 U 1 ) = B(U 1 Λ k ). (Λk+1 Λ k ) = θ k (U k 1 Λ k ) = θ k ( (U k 1 U 1 ) + (U 1 Λ k )) = θ k (B + (U 1 Λ k )) = θ k S (U 1 Λ k ) = θ k (F 1 S Λ k ). Teorem 3. ρ be spectral radius of ( ) 1 S, wic is iterative matrix of preconditioned Ricardson iteration. Ten, tere is a positive constant σ, wic is independent of finite element mes parameter of subdomain, suc tat ρ σ. Teorem 4. Put θ k = θ(k =, 1,,...), ten, tere exists a constant δ( < δ < 1), wic is independent of finite element mes parameter of subdomain. For < θ < δ, te preconditioned Ricardson iteration, i.e., Diriclet-Neumann alternating metod (15)-(16) converges and te convergence rate is independent of mes parameter of subdomain. (Advance online publication: 17 November 17)

4 Proof. From Teorem, we ave U 1 Λ k+1 = (I θ( ) 1 S )(U 1 Λ k ) = (I θ( ) 1 S ) k+1 (U 1 Λ ), it comes tat U 1 Λ k+1 δ k+1 U 1 Λ, δ = I θ( ) 1 S. Following Teorem 3, tere exists a constant δ( < δ < 1), wic is independent of. For < θ < δ, spectral radius of I θ( ) 1 S is less tan 1, and spectral norm δ < 1; terefore, lim U 1 Λ k+1 =. k + It follows tat te preconditioned Ricardson iteration converges; ten, te Diriclet-Neumann alternating metod converges and te convergence rate is independent of mes parameter of subdomain. V. NUMERICAL EXAMPLES In tis section, we give two numerical examples to sow te effectiveness of te Diriclet-Neumann alternating metod. In tese examples, te exact solutions are known. Te purpose of sowing tese examples is to ceck te convergence in terms of iteration k and mes size. Te finite element metod wit liner elements is used in te computation. u 1 is te finite element solution in, e and e denote te maximal error of all node functions in, respectively, i.e., e(k) = sup P i u(p i ) u k 1(P i ), e (k) = sup P i u k+1 1 (P i) u k 1(P i ). q (k) is te approximation of te convergence rate, i.e., q (k) = e (k 1). e (k) Example 1. We consider problem (1), Ω = {(µ, φ) µ > 1, < φ < π}, = {(1, φ) < φ < π}, = {(µ, ) µ > 1}, = {(µ, π) µ > 1} and f =. sin φ cos µ +sin µ u(µ, φ) = be te exact solution of original problem and g = u. µ1 = {(3, φ) < φ < π} be te artificial boundary. Fig. 3 sows te mes of subdomain, Table 1 sows te relation between convergence rate and mes (θ =.5), Table sows te relation between convergence rate and relaxation factor (mes /4), Fig. 4 sows L ( ) errors wit iteration k. Example. We consider problem (1), Ω = {(µ, φ) µ > 1, < φ < 3π }, = {(1, φ) < φ < 3π }, = {(µ, ) µ > 1}, = {(µ, 3π ) µ > 1} and f =. u(µ, φ) = sin φ 3 cos µ 3 +sin µ 3 be te exact solution of original problem and g = u. µ1 = {(3, φ) < φ < π} be te artificial boundaries. Fig. 5 sows te mes of subdomain, Table 3 sows te relation between convergence rate and mes (θ =.6), Table 4 sows te relation between convergence rate and relaxation factor (mes /4), Fig. 6 sows L ( ) errors wit iteration k. Fig. 3. Mes of Subdomain for Example 1 TABLE I THE RELATION BETWEEN CONVERGENCE RATE AND MESH FOR EXAMPLE 1 (θ =.5) Mes k e(k) / e (k) q (k) e(k) /4 e (k) q (k) e(k) /8 e (k) q (k) TABLE II THE RELATION BETWEEN CONVERGENCE RATE AND RELAXATION FACTOR FOR EXAMPLE 1 (MESH /4) θ k e(k) e (k) q (k) e(k) e (k) q (k) e(k) e (k) q (k) e(k) e (k) q (k) Error / /4 / Iteration k Fig. 4. L ( ) Errors wit Iteration k for Example 1 Te numerical results sow tat tis metod is feasible and convergent quickly. Its convergence rate is independent of finite element mes parameter. Te convergence of te metod is te best wen te relaxation factor θ k approaces to.5. (Advance online publication: 17 November 17)

5 Fig. 5. Mes of Subdomain for Example TABLE III THE RELATION BETWEEN CONVERGENCE RATE AND MESH FOR EXAMPLE (θ =.6) Mes k e(k) / e (k) q (k) e(k) /4 e (k) q (k) e(k) /8 e (k) q (k) TABLE IV THE RELATION BETWEEN CONVERGENCE RATE AND RELAXATION FACTOR FOR EXAMPLE (MESH /4) θ k e(k) e (k) q (k) e(k) e (k) q (k) e(k) e (k) q (k) e(k) e (k) q (k) / /4 /8 REFERENCES [1] H. Han and X. Wu, Approximation of infinite boundary condition and its application to finite element metods, Journal of Computational Matematics, vol. 3, no., pp , [] H. Han and X. Wu, Te artificial boundary metod numerical solutions of partial differential equations on unbounded domains. Beijing: Tsingua University Press, 9. [3] K. Feng, Finite element metod and natural boundary reduction, in Proceedings of International Congress Matematicians, 1983, pp [4] K. Feng and D. Yu, Canonical integral equations of elliptic boundary value problems and teir numerical solutions, in Proceedings of Cina- France Symposium on te Finite Element Metods, 1983, pp [5] D. Yu, Coupling canonical boundary element metod wit FEM to solve armonic problem over cracked domain, Journal of Computational Matematics, vol. 1, no. 3, pp. 195-, [6] D. Yu, Approximation of boundary conditions at infinity for a armonic equation, Journal of Computational Matematics, vol. 3, no. 3, pp. 19-7, [7] D. Yu, Natural Boundary Integral Metod and Its Applications. Massacusetts: Kluwer Academic Publisers,. [8] J. B. Keller and D. Givoli, Exact non-reflecting boundary conditions, Journal of Computational Pysics, vol. 8, no. 1, pp , [9] M. J. Grote and J. B. Keller, On non-reflecting boundary conditions, Journal of Computational Pysics, vol. 1, no., pp , [1] D. Yu, A domain decomposition metod based on te natural boundary reduction over an unbounded domain, Matematica Numerica Sinica, vol. 16, no. 4, pp , [11] D. Yu, Discretization of non-overlapping domain decomposition metod for unbounded domains and its convergence, Matematica Numerica Sinica, vol. 18, no. 3, pp , [1] Q. Du and D. Yu, A domain decomposition metod based on natural boundary reduction for nonlinear time-dependent exterior wave problems, Computing, vol. 68, no., pp ,. [13] Q. Du and M. Zang, A non-overlapping domain decomposition algoritm based on te natural boundary reduction for wave equations in an unbounded domain, Numerical Matematics, vol. 13, no., pp , 4. [14] M. Yang and Q. Du, A Scwarz alternating algoritm for elliptic boundary value problems in an infinite domain wit a concave angle, Applied Matematics and Computation, vol. 159, no. 1, pp. 199-, 4. [15] B. Liu and Q. Du, Diriclet-Neumann alternating algoritm for an exterior anisotropic quasilinear elliptic problem, Applications of Matematics, vol. 59, no. 3, pp , 14. [16] Q. Cen, B. Liu and Q. Du, A D-N alternating algoritm for solving 3D exterior Helmoltz problems, Matematical Problems in Engineering, vol. 14, Article ID 41846, 14. [17] X. Luo, Q. Du and L. Liu, A D-N alternating algoritm for exterior 3-D Poisson problem wit prolatesperoid boundary, Applied Matematics and Computation, vol. 69, pp. 5-64, 15. [18] Y. Cen, and Q. Du, Solution of exterior problems using elliptical arc artificial boundary, Engineering ters, vol. 4, no., pp. -6, 16. [19] Y. Cen, and Q. Du, Artificial boundary metod for anisotropic problems in an unbounded domain wit a concave angle, IAENG International Journal of Applied Matematics, vol. 46, no. 4, pp. 6-65, 16. [] Y. Cen, and Q. Du, An iteration metod using elliptical arc artificial boundary for exterior problems, IAENG International Journal of Applied Matematics, vol. 47, no., pp , 17. Error Interation k Fig. 6. L ( ) Errors wit Different Iteration k for Example ACKNOWLEDGMENT Te autors would like to tank te reviewers for teir valuable comments wic improve te paper. (Advance online publication: 17 November 17)