Solving Negative Order Equations by the Multigrid Method Via Variable Substitution

J Sci Comput (214) 59:371 385 DOI 1.17/s1915-13-9762-4 Solving Negative Order Equations by the Multigrid Method Via Variable Substitution George C. Hsiao Liwei Xu Shangyou Zhang Received: 26 December 212 / Revised: 12 July 213 / Accepted: 26 July 213 / Published online: 11 August 213 Springer Science+Business Media New York 213 Abstract Variable substitutions are introduced to the single layer potential equations such that the order of pseudo-differential operator is changed from minus one to plus one. Though the condition number remains the same order after such a variable substitution, the frequencies of higher and lower eigenfunctions are switched. The multigrid iteration is shown to be an optimal order solver for the resulting linear systems of boundary element equations. Two types of variable substitutions are suggested. Numerical tests are presented showing efficiency of both methods, and supporting the theory. Keywords Boundary element Single layer potential First kind integral equation Negative order pseudo-differential operator Multigrid method Mathematics Subject Classification 65N3 65N38 1 Introduction The multigrid method provides an optimal order solver for large linear systems arising from discretizing partial differential equations (cf. [1] and references in [3]). The multigrid method G. C. Hsiao S. Zhang (B) Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA e-mail: szhang@udel.edu G. C. Hsiao e-mail: hsiao@math.udel.edu L. Xu College of Mathematics and Statistics, Chongqing University, Chongqing 41331, People s Republic of China L. Xu Institute of Scientific and Engineering Computing, Chongqing University, Chongqing 444, People s Republic of China e-mail: xul@cqu.edu.cn

372 J Sci Comput (214) 59:371 385 is based on two principles. One is to reduce oscillatory, high frequency components of the iterative error by fine-level smoothings. This is in common for all iterative methods, which are all based on the residual correction, as the high frequency error generates larger residue (in positive order differential equations). The other principle, the key principle of multigrid method, is to reduce the smooth, low frequency components of the error by the coarselevel correction, with the optimal order of computation. However, the multigrid method works only for the positive order elliptic equations. In the pseudo-differential equations of negative orders, the eigenfunctions associated with large eigenvalues are smooth while the eigenfunctions for small eigenvalues are oscillatory. The slow convergent component (high oscillatory) of iterative error can not be corrected by the coarse level solutions. Therefore both the fine-level relaxation and the coarse-level correction could not reduce effectively such low eigenvalue components. Bramble et al. proposed a multigrid method in [3]whereH 1 equivalent, discrete innerproducts are introduced to the fine-level relaxation. It changes the relationship between eigenvalues and eigenvectors of the fine-level smoothing operator such that highly oscillatory components of error can be reduced rapidly. They showed that the multigrid method can converge in a rate independent of the size of the linear system to be solved. However, each evaluation of an H 1 discrete inner-product requires essentially a solution of the Laplace equation (preconditioning the equation under consideration), though the latter can be obtained by an optimal order of computation by the multigrid method. In this paper, we solve boundary element equations of negative order one, arising from exterior boundary value problems, for example, an exterior Dirichlet boundary value problem for the Laplacian in R 2 or R 3, shown in Fig. 1. The condition number for the simple-layer boundary operator is O(h 1 ) (cf. [6]) which is studied in Bramble et al. [3]. However, it is not clear how to construct and to evaluate the H 1 discrete inner-product on the boundary. If the boundary is not smooth, for example, with corners, the surface Laplace equation may not be well defined. Even if the boundary is smooth, solving a surface Laplace equation might be more challenging than solving a surface integral equation itself. Therefore, we seek a different approach in the multigrid method. Instead of changing the order of eigenfunctions in the fine-level relaxation as in [3], we change the order of the differential equation from minus one to plus one, by variable substitutions. For the model problem (2.1) shown in Fig. 1, after introducing the fundamental solution, it is converted to the following weak variational problem: Find σ H, such that V σ, χ = f,χ χ H, (1.1) where f := f /2 + Kf, H, V, K are defined in (2.7), (2.5) and(2.4), respectively, and the unkonwn dnsity function σ = u + / n, the normal derivative of u with respect to the unit normal n shown in Fig. 1. We propose two types of variable substitutions for the unknown density function σ of the single layer potential Eq. (1.1): σ = ẇ in 2D, (1.2) σ = W w in 2D and 3D, (1.3) Fig. 1 An exterior boundary value problem

J Sci Comput (214) 59:371 385 373 where w is an unknown function on,andẇ denotes the tangential derivative along a boundary curve, and W is the hypersingular operator to be defined in (2.14). The first substitution works only for 2D boundary value problems, while the second works for both 2D and 3D problems. We then discretize the positive order equations by continuous boundary elements: Find w h H h (defined by (2.15)), such that or, in the second method, V ẇ h, v h = f, v h v h H h, (1.4) VWw h, W v h = f, W v h v h H h. (1.5) We show that the standard multigrid method solves Eqs. (1.4) and(1.5) in optimal order of computation, i.e., obtaining an N-dimensional iterative solution up to the truncation error by an order N computation, [1]. Basically, it only requires to show the convergence rate of the multigrid method is constant, independent of the grid size h, or the size N of the linear system to be solved. In the first case, we show the discrete solution obtained by solving (1.4) is exactly the solution of original boundary integral equation: σ h = ẇ h, where σ h is the solution of (2.12). But in the second case, (the conclusion here may depend on the implementation methods,) we obtain another discrete solution in a different function space, σ h = W k w h σ h. It is not clear how to choose numerical quadratures for (1.5), so that the new solution w h would produce the same solution σ h = W w h of (1.5), either at the continuous level, or at the discrete level. We choose a specific numerical integration method in computing (1.5). The discrete solution W k w h converges at the third order, one order higher than the optimal order h 2 of the piecewise constant function space. In our computation, we made w h H h, the space of continuous linear functions on the dual grid. But the optimal order is still O(h 2 ) in L 2 -norm for the piecewise constant function σ h = W h w h. We get a super-convergence (numerically only for four examples) in the second method. The second variable substitution method proposes a new discretization method. The super-convergence analysis of the method is to be done in another paper while this work is limited to the multigrid convergence of the method. In practice, we recommend the first method in 2D, (1.4), where with almost no additional cost, the computation is brought down from O(h 2 log h) to the optimal order O(h 1 log h) (assuming a fast evaluation method such as FFT is used for the bilinear form). But in 3D, we are limited to the second variable substitution method, (1.5). The rest paper is organized as follows. In Sect. 2, we introduce the problem, a boundary element discretization, and variable substitutions. In Sect. 3, we show the constant rate of convergence for the multigrid method when solving the substituted equations. In Sect. 4, we show that the solutions of the original negative one order equation and of the substituted positive one order equation are same, for the d/ds substitution. In Sect. 5, we present some numerical results for W substitution.

374 J Sci Comput (214) 59:371 385 2 Variable Substitution We consider the following model exterior Dirichlet problem (cf. Fig. 1) in2dor3d: u = in c := R d \, u = f on :=, u = O(1) or O( x 1 ) as x (2.1) depending on the dimension d = 2or3,where f H 1/2 ( ) is a given function. Here is assumed to be a smooth domain, but we assume there is no curve approximation error and no quadrature error. We use the standard Sobolev space notations (cf. [4,7]). By the Green s formula, we begin with the representation of the solution of (2.1) inthe form: u(x) = u + (y) γ (x, y) ds y n y γ(x, y) u+ n y (y) ds y + c (2.2) for all x c,wherec = in 3D, or in 2D c is an unknown constant depending on f and the boundary shape of in the boundary condition of (2.1), and n is the normal vector to. In 2D, the constant c will be computed by (2.8) below, after u + / n is computed. Here the fundamental solution of the Laplacian is defined by { ( 1/2π)log x y if d = 2, γ(x, y) = 1/(4π x y ) if d = 3. Let x approach from outside, i.e., u(x) = f (x) on. It follows that 1 2 f (x) = f (y) γ (x, y) ds y γ(x, y) u+ (y) ds y + c x, n y n y and that u + (y) n y ds y = in 2D. Representing u + / n by σ, the above equation is, ( ) 1 V σ c = 2 I K f on, (2.3) where K v(x) = γ(x, y)v(y) ds y n y v H 1/2 (), (2.4) V σ(x) = γ(x, y)σ (y) ds y σ H 1/2 (), (2.5) and σ is required to satisfy the condition σ ds =, (2.6)

J Sci Comput (214) 59:371 385 375 in the case of d = 2. In order to simplify the presentation, in the following let us denote the space { H 1/2 () in 3D, H = H 1/2 () ={χ H 1/2 (2.7) () 1,χ =} in 2D, and introduce the notation v, χ = vχ ds v H 1/2 (), χ H. Multiplying the Eq. (2.3) by the test functions χ, we get a variational problem for (2.1), i.e., V σ, χ = f,χ χ H, where f = ( 2 1 I K ) f. In the case of d = 2, the unknown constant c in the solution (2.2) is determined by c = (V σ + ((1/2)I K ) f )ds ds. (2.8) In this paper, we confine our presentation in 2D. To define boundary element spaces, we first subdivide the boundary ={y(s) s S(arc length)} into a quasi-uniform segments of curves, to obtain an initial grid 1 ={E 1, j ={y(s) s j 1 s s j }, j = 1, 2,...,N 1 }. (2.9) The multigrids { k } on are defined sequentially by cutting previous elements into halves: k ={E k, j E k 1, j = E k,2 j 1 E k,2 j, E k 1, j k 1 }. (2.1) Let h k be the mesh size of k, h k = max E k, j, 1 j N k where N k = N 1 2 k 1,and E k, j is the arc length of curve E k, j. We note that h k = h k 1 /2. Let the boundary element spaces be the piecewise constant spaces on the multigrids k, S hk ={v H 1/2 () v Ek, j P, j = 1, 2,...N k }. (2.11) We note that dim S hk = N k 1. The boundary element discretization for the single layer potential equation can be formulated: Find σ hk S hk such that V σ hk,χ = f,χ χ S hk. (2.12) Let {φ j } be nodal basis functions of S hk, i.e., φ j = 1onE k, j but on rest elements of k. The problem (2.12) can be written in a matrix form, B k σ hk = f hk (2.13)

376 J Sci Comput (214) 59:371 385 Table 1 The errors and the iteration numbers, for (2.12) with (2.17) k σ σ hk L 2 h n k #jc σ σ hk L 2 h n k #jc C(B k ) a = 1in(2.17) a = 1.9in(2.17) 1.34393554. 11 1.1763686. 12 1.5 2.1247276. 42 1.7548997. 49 2.89 3.43419451 1.2 83 1.66872981.1 112 5.77 4.11132593 2. 128 1.11283268.6 231 11.54 5.2731438 2. 161.21629389 2.4 457 23.8 6.67479 2. 183.28499361. 876 46.16 7.167626 2. 198.11325364 1.3 1,556 92.33 8.41769 2. 212.2829578 2. 2,374 184.66 Fig. 2 Eigenvectors v 2 (for smallest eigenvalue) and v Nk,cf.(2.18) where the entries of matrix B k, vectors σ hk and f hk are defined respectively, by (B k ) i, j = Vφ j,φ i, ( σ hk ) j = c j so that σ hk = c j φ j, 1 j N k ( f hk ) j = f,φ j. When solving the linear system of Eq. (2.13), the rate of convergence of the iterative solution depends on the condition number of B k, for the usual iterative methods such as the conjugate gradient method or the Jacobi method. The condition number of B k is of O(h 1 k ), as shown numerically in Table 1. The standard multigrid method would not accelerate these iterations either, as the slow convergence component of the iterate is highly oscillatory, which can not be approximated by the coarse level solutions. We can see this in Fig. 2 which is obtained from (2.18). Next, we introduce variable substitutions by (1.2)and(1.3), i.e., σ = ẇ = d w, σ = W w. ds Here W is a hypersingular operator defined by W w(x) = γ (x, y)w(y)ds y, x. (2.14) n x n y

J Sci Comput (214) 59:371 385 377 Fig. 3 The solution σ hk for (2.12) with a = 1.9in(2.17) We next introduce the boundary element spaces, H hk ={v H 1/2 () C () v Ek, j P 1 E k, j k }, (2.15) consisting of continuous, piecewise linear functions on grid k. We note that for v H hk,v Ek, j is a linear function of arc length, i.e, it is a linear function on the reference element. On E k, j, its graph is a space curve, not on a plane. The two spaces S hk and H hk are of the same dimension, dim H hk = dim S hk = N k 1. Let {ψ j } be nodal basis function of H hk. Then we represent both linear systems for (1.4)and (1.5)by A k w hk =ḡ hk, (2.16) where the matrix and vectors are defined respectively, by (A k ) i, j = V ψ j, ψ i or VWψ j, W ψ i, N k ( w hk ) j = w j so that w j ψ j = w hk, j=1 (ḡ hk ) j = f, ψ j or f, W ψ j. We will discuss the multigrid method for solving the substituted Eq. (2.16)in the next section. To conclude this section, we show the necessity of variable substitution for the multigrid method, by using a simple numerical example. Let be the circle of radius 8 centered at the origin. Let the exact solution of (2.1)be u = log (x a)2 + (y a) 2 (x a) 2, a < 2. (2.17) + (y + a) 2 Now let be cut by the four points (±2, ±2) to be the initial grid 1. We solve the discretized problem (2.12) by the Jacobi iteration. In Table 1, we list the errors in L 2 norm of solutions σ hk, the convergence order, and the number of Jacobi iterations used to solve the boundary element equations with a = 1and1.9 in(2.17). We note that when a 2, the solution in (2.17) is nearly singular. The solution σ hk when a = 1.9 is plotted in Fig. 3.

378 J Sci Comput (214) 59:371 385 In Table 1, #jc stands for the number of Jacobi iterations. We also list in Table 1 the condition number C(B k ) of matrices B k in (2.13). We note that the first eigenvalue of B k is negative, whose associated eigenvectors are a global constant, which is not in the function space S hk. The rest eigenvalues are positive, numbered by Numerically, we have B k v j = λ k, j v j, <λ k,2 λ k,3 λ k,nk. (2.18) λ k,2 h k, λ k,nk 1. But the eigenvector v 2 is oscillatory, while v Nk is smooth, shown in Fig. 2. The (effective) condition number of B k is defined by C(B k ) = λ k,n k λ k,2 = O(h 1 k ). The numerical values of the condition number C(B k ) are listed in the last column of Table 1. 3 The Multigrid Method We define a standard multigrid method for solving the positive order boundary element Eq. (2.16). We represent both variable substitutions by the form Aw, v = Vẇ, v or VWw, W v w, v H 1/2. (3.1) We may simply write the derived operator A as A = d ds V d ds or WVW. We note that W is symmetric. We remark that (cf. [7,8]) V μ, ν = Wμ, ν μ, ν H 1/2 (). Now, as W : H 1/2 () H 1/2 () is a pseudo-differential operator of order plus one, for both cases in (3.1), the operator A is a pseudo-differential operator of order plus one: A : H 1/2 () H 1/2 (). For analyzing the multigrid method, we introduce the usual discrete norms on the discrete spaces H hk (cf. [1]): w hk k, s = A 2s k w h k,w hk, w hk H hk, s 1. (3.2) Here the symmetric positive-definite operator A k (also used to denote the discrete matrix in (2.16)) is a restriction of A (3.1) on the boundary element spaces: A k : H hk S hk, A k w hk,v hk = Aw hk,v hk w hk,v hk H hk. Similar to [1], we define a multigrid method (recursively) for solving the linear systems (2.16):

J Sci Comput (214) 59:371 385 379 Definition 3.1 (A multigrid method) Given an initial w approximating the solution w hk in (2.16), one k th level multigrid iteration produces a new approximation w m+1 as follows. First, m Richardson smoothings are performed: w l = w l 1 + ρ(a k ) 1 (ḡ k A k w l 1 ) l = 1,...,m, (3.3) where ρ(a k ) stands for the spectral radius of the symmetric positive-definite matrix A k or an upper bound of it. As in (2.16), w denotes the coefficient vector of function w H hk. Then we solve the residual problem on the coarse level A k 1 q,v hk 1 = g k A k w m,v hk 1 v hk 1 H hk 1. (3.4) by doing p( 1)(k 1) st level multigrid iterations to get a solution q,ifk > 1. If k = 1 we solve (3.4) exactly to get q = q. Finally w m+1 = w m + q. (3.5) We note that there is an interpolation operator I k : H hk 1 H hk in the residual Eq. (3.4) and in the coarse level correction (3.5). But as the multi-level spaces are nested, H hk 1 H hk, I k is the identity operator, or natural embedding, I k v hk 1 = v hk 1. But computationally, I k is needed for transferring functions from one grid to the next. Lemma 3.1 The discrete norms are bounded by the fractional order Sobolev norms: for any v hk H hk, Proof First, we have, cf. [7], v hk k,s C v hk H s, s 1 2. (3.6) v H 1/2 W v, v v H 1/2 (). Here we use a notation A B for an equivalence, i.e., C 1 A B C 2 A for some positive constants C 1 and C 2. For any v hk H hk, we have two definitions of the discrete norm, depending on the variable substitution method. For the first case, A k = ds d V ds d,foranyv h k H hk,wehave v hk H 1/2 W v hk,v hk = V v hk, v hk = A k v hk,v hk = v hk k,1/2. For the second case, A k = WVW, we need also to use another norm equivalence: For any v hk H hk,wehave For both cases, we have σ H 1/2 V σ, σ. v hk H 1/2 W v hk,v hk = W 1/2 v hk L 2 W v hk H 1/2 VWv hk, W v hk = A k v hk,v hk = v hk k,1/2. v hk H 1/2 C v hk k,1/2 v hk H = v hk k, = v hk,v hk. Thus, (3.6) is obtained by the interpolation, cf. [4].

38 J Sci Comput (214) 59:371 385 Lemma 3.2 The following smoothing property holds for (3.3): e m k,1/2+α Cm α h α k e k,1/2, <α 1 2 (3.7) where the iterative errors are denoted by e l = w hk w l. Proof The proof is similar to that for the second order differential operator [1]. Here we have a first order pseudo differential operator. Let A k have the following (orthonormal) eigen expansion: A k v j = λ k, j v j, <λ k,2 λ k,nk. Here {v j } forms an orthogonal basis for H hk products. Let e be expanded under the basis with respect to both, and A, inner N k e = c j v j. j=2 By (3.3), Thus, N k ( e m = (I ρ(a k ) 1 A k ) m e = 1 λ ) m k, j c j v j. λ k,nk j=2 j=2 N k e m 2 k,1/2+α = ( 1 λ ) 2m k, j c 2 j λ λ1+2α k, j k,nk ( max (1 x [,1] x)2m x 2α ) λ 2α k,n k N k j=2 c 2 j λ1 k, j = Cm 2α ρ(a k ) 2α e 2 k,1/2 Cm 2α h 2α k e 2 k,1/2, where we have used the standard inverse inequality for estimating ρ(a k ), i.e., ρ(a k ) Ch 1 k. This completes the proof. Theorem 3.1 (Constant-rate convergence for MG) For any positive γ<1, there exist constants m (large enough) and p(> 1), both independent of k, such that Here w hk and w i are defined in Definition 3.1. w hk w m+1 k,1/2 γ w hk w k,1/2. (3.8) Proof We will prove the convergence of two-level multigrid method (i.e., q = q in (3.5)). By which it is standard to show the W-cycle (p > 1) convergence (cf. [1]). The proof here is similartothatin[1], except the notations and the order of the pseudo-differential operator. Let the iterative errors be denoted by e l = w hk w l H hk.

J Sci Comput (214) 59:371 385 381 Noticing the orthogonal projection property of the coarse-level correction (3.4), (3.5), we have e m+1 2 k,1/2 = A k(e m q), e m e m+1 k,1/2 α e m k,1/2+α, (3.9) where <α 1/2isthe order of regularity for the boundary value problem (2.1) inthe following sense. For any Dirichlet data f H 1/2+α () in (1.1), there is a unique solution σ H 1/2+α () in (1.1) such that σ H 1/2+α () C f H 1/2+α (). (3.1) This regularity result is a direct consequence of the elliptic regularity for the original boundary value problem (1.1) after applying the Sobolev trace theorem, cf. [7,8]. We next apply a duality argument to the first term of (3.9). For any g H 1/2+α () with g H α+1/2 () = 1, let σ g be the solution of (1.1) with f there being replaced by g. Wethen let u g be a density function of σ g, i.e., σ g = u g in the first case of (3.1), or σ g = Wu g in the second case. By the equivalence of norms in (3.6), the regularity assumption (3.1) andthe standard approximation properties of boundary elements, we get e m+1 k,1/2 α C e m+1 H 1/2 α () = C ė m+1 H 1/2 α () = C sup g H 1/2+α () =1 ė m+1, g =C sup ė m+1, V σ g g = C sup ė m+1, V u g =Csup A(e m q), u g g g C sup e m q H 1/2 u g I k 1 u g H 1/2 g C e m q k,1/2 sup h α k 1 u g H 1/2+α () g = C e m q k,1/2 sup h α k 1 σ g H 1/2+α () g = C e m q k,1/2 sup h α k 1 g H 1/2+α () g = Ch α k 1 e m+1 k,1/2. (3.11) Here I k 1 is the standard nodal interpolation operator onto H hk. As we assume α>, I k 1 is well defined in 2D. But in 3D, as well as in 2D, it can be replaced by an averaging interpolation operator, Scott-Zhang operator [1]. We remark that for s = and 1/2, (3.6) can be reversed, i.e, v hk H 1/2 C v hk k,1/2. But it is not clear for middle s (though it is widely used). The proof for the second case of (3.1) is same except replacing the dot operator in (3.11)bythe W operator. Therefore, combining (3.9), (3.7) and(3.11), (3.8) is shown when m is chosen large enough such that Cm α γ. 4Thed/ds Substitution When we make two variable substitutions in (3.1), the original PDE solutions are exactly the same as that obtained from the two potential solutions: Au,v = f, v or f, W v, v H 1/2 (). That is, σ = u or σ = Wu.

382 J Sci Comput (214) 59:371 385 But at the discrete level, it is not clear what is the relationship between σ hk of (2.12)andw hk of (2.16). This is to be studied in this section and next, for the two cases. Proposition 4.1 The solution of (2.12) is the tangential derivative of the solution of (2.16), σ hk = ẇ h,k, when A = ds d V ds d, i.e., w h k H hk such that V ẇ hk, v = f, v v H hk. (4.1) d Proof ds is an one-one on-to mapping from H h k to S hk. Equation (4.1) implies V ẇ hk,χ = Vf,χ χ S hk. Its unique solution is then σ hk of (2.12). We note that there is no numerical (truncation) error when computing ẇ hk as the function is a linear function of arc length on each curve segment. In computation, we evaluate Aw hk,v by V ẇ hk, v. There is no numerical error/difference between solving the original Eq. (2.12) and solving the substituted Eq. (4.1). But this may not be true for our second variable substitution, to be studied next section. Theorem 4.1 (Optimal order of computation) By an order O(N hk ) of computation, the multigrid solution w hk ( w hk ) of (4.1) approximates the true solution σ of (1.1) at the optimal order: σ w hk L 2 () Ch 2 f H 1 (). Proof By (3.8) and by choosing the previous level solution as the initial guess, the multigrid method would produce a solution w hk up to the truncation error by a constant number of iterations (independent of the size of the linear systems of equations). This is called an optimal order of computation of the multigrid method, cf. [1]. Then we assume the boundary has some local symmetry so that the matrix B k in (2.13) is locally cyclic. If so, B k σ k can be evaluated in O(N hk ) or O(N hk log N hk ) computation, via the fast Fourier transform method on B k or on its submatrices, cf. [14]. The fast multipole method may also provide an evaluation of A k w hk by order N k operations, [5]. The total work for the multigrid method would be of O(N hk ). We end this section by a numerical test on the multigrid method with d/ds variable substitution. We solve (4.1) with the exact solution from (2.17) with a = 1. The errors by the variable substitution method are listed in the second column of Table 2, which are exactly thesameasthatintable1. This is shown in Proposition 4.1. The linear systems of Eq. (4.1) are solved by three methods, the Richardson iteration (3.3), the conjugate gradient method, and the multigrid method. The number of iteration is listed in Table 2, for all three methods. As the condition number of A k is of order h 1 k, we need O(2 k ) Richardson iterations, O( 2 k ) conjugate gradient iterations, but O(1) multigrid iterations (shown in Theorem 3.1). In our computation, we use the V-cycle multigrid method with four fine-level smoothings, i.e., p = 1andm = 4 in Definition 3.1. Here the computation is done on a Toshiba Satellite A135-S4656 laptop with a 1.6GHz Intel Celeron M Processor 52 and a memory of 512MB. We listed the computation time for each computation in Table 2. The growth rate of computation time for the multigrid method is 4, instead of 2, predicted by Theorem 4.1. This is because we did not use the fast Fourier transform to evaluate A k w hk, which costs us O(N 2 k ) operations instead of O(N k) operations. Also we can use the fast multipole method to evaluate A k w hk,byanordern k operations, [5].

J Sci Comput (214) 59:371 385 383 Table 2 The errors, iterations, and CPU time, for (4.1) with (2.17) k σ w hk L 2 h n k #jc cpu #cg cpu #mg cpu 1.343935. 18 2 1 2.12472. 4 4 6 3.434194 1.2 86 7 6 4.111325 2. 178 13 6 5.27314 2. 363 1 19 6 6.6747 2. 733 7 24 5 7.1676 2. 1,474 47 32 1 5 1 8.417 2. 2,955 345 44 6 5 4 9.14 2. 5,916 2,622 61 26 5 14 1.26 2. 85 145 5 54 5TheW Substitution The substitution method σ hk = ẇ hk works well in 2D. But it cannot be extended to 3D. The other substitution method σ hk = W w hk would work in both 2D and 3D. Our early analysis remains the same in 3D. However, computationally, it is not fully clear yet how to implement the method: Find w hk H hk such that VWw hk, W v hk = f, W v hk v hk H hk. (5.1) The central problem is on the evaluation of hypersingular operator W, especially in 3D. There are many methods, but all are complicated, cf. [11] and references therein. In our computation, we use the 2D formula W w, v = Vẇ, v w, v H 1/2 (). This way, the hypersingular operator is evaluated via a weakly singular operator. This is a widely used approach. Therefore, the cost of evaluating a matrix vector product A k w in the W substitution method is about three times of that A k w evaluationinthed/ds substitution method. But there is still another problem, finding appropriate quadrature formulas for (5.1). To evaluate V ẇ hk accurately, when ẇ hk has a jump at each grid points, we introduce a dual grid space. In other words, it is natural and accurate numerically, to find the mapping V : S hk H 1/2 () H hk H 1/2 (), where H hk is defined the same way as H hk butonadualgrid k of k. That is, the grid points of k are the mid-points of elements of k. This is perfect for evaluating, in our last method of ẇ-substitution, V ẇ hk, v hk w hk,v hk H hk, where one mid-point quadrature formula is used for the boundary integral. At a middle point, ẇ hk is continuous. Therefore, we introduce an approximation of W operator: W k : H hk S hk, (5.2) W k w hk,v hk = V k ẇ hk, v hk, (5.3)

384 J Sci Comput (214) 59:371 385 Table 3 The errors, iterations, and CPU time, for (5.1) with a = 1in(2.17) k I k σ W w hk L 2 h n k #cg cpu #mg cpu 1.1488936787. 3 1 2.13612467316 3.4 4 1 3.12448759497.1 8 1 4.15444779648 3. 13 2 5.1748794751 3.1 17 3 6.212825341 3. 18 4 2 7.2641945 3. 21 2 5 6 8.3296587 3. 19 6 8 31 9.411888 3. 22 25 14 25 where V k : S hk H hk, (5.4) V k v hk Ek, j = (V v hk )(x k, j ) E k, j k. (5.5) In (5.5), x k, j is the middle point of a curve segment E k, j. By this method, we change the Eq. (5.1)to V k W k w hk, W k ṽ hk = f, W k v hk v hk H hk. (5.6) In Table 3, we list the error and the number of iterations when solving the linear system (5.1) with exact solution (2.17) with a = 1. Because the coarse level solution of the residual problem (3.4) is no longer an projection of the iterative error, we have a nonnested multigrid method, due to the perturbation of W k to W in (5.3). The convergence of nonnested multigrid methods is not guaranteed, cf. [2,12,13,15]. Extra steps of fine-level smoothings (3.3) are needed to suppress the nonnested error from the coarse level correction. Here, we change the Richardson iteration (3.3) in the multigrid method (Definition 3.1) to the conjugate gradient iteration. We note that the conjugate gradient iteration reduces errors of all frequencies. Here we use V-cycle multigrid method with eight smoothings, (p = 1andm = 8 in Definition 3.1). This barely makes the nonnested multigrid method converge. From the CPU time used in Table 3, this time the multigrid method does not beat the conjugate gradient method yet (cf. Table 2). Of course, when the level is low, i.e., for small linear systems, the conjugate gradient method would be always faster than the multigrid method (of optimal order). In 2D, the second substitution, the W substitution, seems much worse than the d/ds substitution, from the computational cost. However, numerically, we found a superconvergence for the W substitution. In Table 3, we can see that the order of convergence is 3, one order higher than the optimal order, 2. Where in Table 3 I k stands for the mid-point interpolation operator from H 1 () to S hk. We do not have a proof for this phenomenon yet. We further tested a few cases, i.e., the exact u in (2.1)is: 3 ( ) r ni u(r,θ) = cos(n i θ), (5.7) 8 i=1 u(x, y) = e x sin y (an interior solution.) (5.8)

J Sci Comput (214) 59:371 385 385 Table 4 The error e k = I k σ W w hk of (5.1) with various exact solutions k e k L 2 h n k e k L 2 h n k e k L 2 h n k (2.17), a = 1.5 (5.7), n i = 3, 7, 11 (5.8) 1.6583965. 65.27638. 2.6217962. 2.43658577.6 41.679687.6 8.2667578. 3.4266491 3.4 46.718437. 1.31247788 2.7 4.27473.6 1.273395 2.2.11484485 3.5 5.44827 2.6.96812 3.4.129617 3.2 6.49414 3.2.1132 3.1.156725 3. 7.5954 3.1.1342 3..19443 3. 8.737 3..1663 3..2425 3. 9.91 3..27 3..33 3. In all cases we tested, we get a third order convergence. Three more test results are listed in Table 4. Again, the convergence of the W -substitution method (due to numerical integration methods) is one order higher than that of standard boundary element method (cf. Table 2). Acknowledgments Liwei Xu is partially supported by the Youth 1 Plan start-up grant of Chongqing University (No. 28114413) in China. References 1. Bank, R., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput. 36(153), 35 51 (1981) 2. Bramble, J.H., Pasciak, J.E., Xu, J.: The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems. Math. Comput. 51(184), 389 414 (1988) 3. Bramble, J.H., Leyk, Z., Pasciak, J.E.: The analysis of multigrid algorithms for pseudodifferential operators of order minus one. Math. Comput. 63(28), 461 478 (1994) 4. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994) 5. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325 348 (1987) 6. Hsiao, G.C., Wendland, Q.L.: Boundary Element Method: Foundation and Error Analysis. Chapter 12 in Encyclopedia of Computational Mechanics, edited by Erwin Stein et al. vol. 1, Wiley (24) 7. Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Applied Mathematical Sciences, 164. Springer, Berlin (28) 8. Hsiao, G.C.: On boundary integral equations of the first kind. J. Comput. Math. 7, 121 131 (1989) 9. Hsiao, G.C., Zhang, S.: Optimal order multigrid methods for solving exterior boundary value problems. SIAM J. Numer. Anal. 31(3), 68 694 (1994) 1. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(19), 483 493 (199) 11. Yu, D.H.: Natural Boundary Integral Method and Its Applications. Translated from the 1993 Chinese Original. Mathematics and its Applications, 539. Kluwer, Dordrecht; Science Press, Beijing (22) 12. Zhang, S.: Optimal-order nonnested multigrid methods for solving finite element equations. I. On quasiuniform meshes. Math. Comput. 55(191), 23 36 (199) 13. Zhang, S.: Optimal-order nonnested multigrid methods for solving finite element equations. II. On nonquasiuniform meshes. Math. Comput. 55(192), 439 45 (199) 14. Zhang, S.: On the convergence of spectral multigrid methods for solving periodic problems. Calcolo 28(3 4), 185 23 (1991) 15. Zhang, S.: Optimal-order nonnested multigrid methods for solving finite element equations. III. On degenerate meshes. Math. Comput. 64(29), 23 49 (1995)