Journal of Computational and Applied Mathematics. The Trefftz method using fundamental solutions for biharmonic equations

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1 Journal of Computational and Applied Mathematics 35 (0) Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: wwwelseviercom/locate/cam The Trefftz method using fundamental solutions for biharmonic equations Zi-Cai Li a,b, Ming-Gong Lee c, John Y Chiang b,, Ya Ping Liu d a Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 8044, Taiwan b Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, 8044, Taiwan c Department of Applied Statistics, Chung-Hua University, HsinChu, Taiwan d Mathematical College, Sichuan University, Chengdu, 60064, China a r t i c l e i n f o a b s t r a c t Article history: eceived 8 September 009 eceived in revised form 3 February 0 Keywords: Method of fundamental solutions Almansi s fundamental solutions Biharmonic equation Singularity problem Trefttz method Error and stability analysis In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations The bounds of errors are derived for the MFS with Almansi s fundamental solutions (denoted as the MAFS) in bounded simply connected domains The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively The stability analysis of the MAFS is also made for circular domains Numerical experiments are carried out for both smooth and singularity problems The numerical results coincide with the theoretical analysis made When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems Moreover, the numerical solutions by the MAFS with Almansi s FS are slightly better in accuracy and stability than those by the traditional MFS Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity 0 Elsevier BV All rights reserved Description of MFS For simplicity, first consider the homogeneous biharmonic equation with the clamped boundary conditions u = 0 in S, u = f on Γ, u ν = g on Γ, () () (3) where = x + = u is the outward normal derivative to Γ, Γ ν, S is the bounded simply connected domain, u y ν is its boundary, and f and g are the known functions smooth enough In real application, we may encounter the nonhomogeneous equation u = p(x, y) in S Suppose that a particular solution ū is found so that ū = p(x, y) in S By means of a transformation w = u ū we have the homogeneous biharmonic equation w = 0 in S with the clamped Corresponding author address: chiang@csensysuedutw (JY Chiang) /$ see front matter 0 Elsevier BV All rights reserved doi:006/jcam00304

2 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) boundary conditions w = f = f ū on Γ and wν = ḡ = g ū ν on Γ Hence we may simply consider () (3) The general solutions of biharmonic equations can be represented by u = u(ρ, θ) = ρ v + z, where (ρ, θ) are the polar coordinates, and v and z are the harmonic functions Denote r = PQ, P = ρe iθ, Q = e iϕ, ϕ is a radian with 0 ϕ π, and i = Then r = + ρ ρ cos(θ ϕ) Hence the fundamental solutions of biharmonic equations in D are found from (4) as Φ(ρ, θ) = r ln r = ( + ρ ρ cos(θ ϕ)) ln + ρ ρ cos(θ ϕ) (5) Denote Φ j (ρ, θ) = r j ln r j, φ j (ρ, θ) = ln r j, where r j = PQ j = + ρ ρ cos(θ ϕ j ) and Q j = e iϕ j with ϕ j [0, π] Hence we may choose the linear combinations of (6) and (7): v N = cj Φ j (ρ, θ) + d j φ j (ρ, θ), where c j and d j are the unknown coefficients to be determined by the boundary conditions () and (3) We may use the Trefftz method [] Denote V N the set of (8) Then the Trefftz solution u N is obtained by I(u N ) = min v V N I(v), where the energy I(v) = (v f ) + w (v ν g), (0) Γ Γ ν is the normal of Γ, and w is the weight chosen as w = /N in computation Almansi s fundamental solutions (simply denoted Almansi s FS) for biharmonic equations are obtained directly from (4) Then Almansi s FS is given by Φ A (ρ, θ) = ρ ln + ρ ρ cos(θ ϕ), () while the fundamental solutions (5) are called the traditional FS in this paper We may choose the linear combination of () and (7): v A = N cj Φ A (ρ, θ) + j d jφ j (ρ, θ), to replace (8), where (4) (6) (7) (8) (9) () Φ A (ρ, θ) = j ρ ln r j = ρ ln + ρ ρ cos(θ ϕ j ) (3) The coefficients c j and d j can also be obtained from the Trefftz method (9) For the biharmonic equation, the MFS and numerical experiments are carried out for (8) and () in [ 5] The other kind of fundamental solution is also introduced in [] Next, let us consider the mixed type of the clamped and simply support boundary conditions on Γ Then Eq (3) is replaced by (see [6]) u ν = g on Γ, u νν = g on Γ, where Γ Γ = Γ, and Γ Γ = The admissible functions (8) and () remain, but the energy I(v) in (9) is replaced by I (v) = (v f ) + w (v ν g) + (w ) (v νν g ), Γ Γ (5) Γ where the weight w = w = /N (4)

3 435 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) Below, we take the FS in (8) for example, and formulate the collocation equations from (4) For Almansi s FS in (), the formulation of collocation equations is similar We have u N = u N (ρ, θ) = c j (r j ln r j ) + d j ln r j = {c j Φ j (ρ, θ) + d j φ j (ρ, θ)}, (6) where c j and d j are the coefficients, and r j = PQ j By following [7], we choose the uniform collocation nodes on an enlarged circle of S : Q k = (, ϕ k ), ϕ k = π k Then we obtain the collocation equations for the mixed type of boundary conditions: N u N (ρ k, θ k ) = c j Φ j (ρ k, θ k ) + w ν u N(ρ k, θ k ) = w w ν u N(ρ k, θ k ) = w d j φ j (ρ k, θ k ) = f (ρ k, θ k ), (ρ k, θ k ) Γ, (7) c j ν Φ j(ρ k, θ k ) + w c j ν Φ j(ρ k, θ k ) + w d j ν φ j(ρ k, θ k ) = wg(ρ k, θ k ), (ρ k, θ k ) Γ, (8) d j ν φ j(ρ k, θ k ) = w g (ρ k, θ k ), (ρ k, θ k ) Γ, (9) where ν is the normal of Γ and Γ, and w = /N This paper is organized as follows In Section, the error bounds are derived for the MAFS with Almansi s FS, and in Section 3, the stability analysis of the MAFS is also made for circular domains In Sections 4 and 5, numerical experiments are carried out for the smooth and singular problems, respectively In the last section, a few remarks are made Error analysis for the MAFS with Almansi s FS In this section, from [7,8] we will develop the error analysis of the MAFS with Almansi s FS for biharmonic equations with the clamped boundary conditions () and (3) Denote two harmonic polynomials of degree n, P h (ρ, θ) = a n 0 n + ρ i (a i cos iθ + b i sin iθ), () i= n P H (ρ, θ) = a 0 n + ρ i (a i cos iθ + b i sin iθ), () i= with the coefficients a i, b i, a i u n = P A n + A n, and b i The biharmonic solutions can be denoted by where the biharmonic polynomials of degree n + and the residuals are given by P A = n P A (ρ, θ) = n ρ P H (ρ, θ) + n P h n (ρ, θ), (4) A = n A(ρ, θ) = n ρ H (ρ, θ) + n h n (ρ, θ), (5) respectively The boundary norm is defined by / v B = v 0,Γ + v w ν, 0,Γ and the Sobolev norms in H k (S) are defined by / k v k = v k,s = v l,s l=0 Suppose that the solution has the regularity property, u H p (S), p 3 There exist the bounds for the residuals, H n (ρ, θ) k,s, h n (ρ, θ) k,s C (3) (6) (7) (8) n p k u p,s, k = 0, (9)

4 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) There exist the bounds for the function v satisfying v = 0, v k,γ C v k+,s, v ν k,γ C v k+ 3,S, (0) where C is a constant independent of v Hence from w = and N N n in computation, we have A n B A n 0,Γ + w C{ A n +,S w A n 3,S} C u p,s () From (0) we have ν A n 0,Γ n p u u A N B = inf v V N u v B () Let v = ū A N, where ūa N := Σ N(P A n ; ρ, θ) is a special linear combination of () using Almansi s FS, to approximate the biharmoic polynomial P A n of degree n + Hence we obtain u u A N B u ū A N B P A n ūa N B + u P A n B = P A Σ n N(P A ; ρ, θ) n B + A n B (3) Since the bounds of A n B are given in (), the important work is to find the bounds of P A Σ n N(P A; ρ, θ) n B Since 0 < ρ 0 ρ C in Γ, (4) the errors P A Σ n N(P A ; ρ, θ) n B, (5) have, essentiality, the same bounds P h n Σ N(P h n ; ρ, θ) 0,Γ, for Laplace s equations Let h = π and Choose the following two special linear combinations [8], N (6) v h = Σ N N(P h ; ρ, θ) = n d h φ k k(ρ, θ), k= v H = Σ N N(P H ; ρ, θ) = n d H φ k k(ρ, θ), k= where the coefficients are given explicitly by and d h k = α k,0a 0 d H k = α k,0a 0 α k,0 = + + h π ln, β k,m = h mm π n (α k,m a m + β k,m b m ), m= n (α k,m a + β m k,mb ), m m= (7) (8) (9) (0) α k,m = h m m cos mkπ, m =,,, () π sin mkπ, m =,, () By following Li [7,8], we have the following lemma Lemma Let (4) hold and N satisfy N q+ (3) The notation N n or N O(n) denotes that there exist two constants C and C such that C n N C n

5 4354 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) For P H n and P h n there exist the bounds, P H n Σ N(P H n ; ρ, θ) q,γ CN q ν {P H Σ n N(P H n ; ρ, θ)} q,γ r max n N rmax CN q+ r max n P H n 0,Γ, (4) n N rmax where r max = max r Γ, = min r Γ and C is a constant independent of N and n Lemma Let (4) hold There exist the bounds, n P H n 0,Γ, (5) ρ P H Σ n N(ρ P H ; ρ, θ) n k,γ C P H Σ n N(P H ; ρ, θ) n k,γ, (6) ν {ρ P H Σ n N(ρ P H n ; ρ, θ)} C P H Σ n N(P H ; ρ, θ) n k+,γ + ν {P H Σ n N(P H n ; ρ, θ)}, (7) k,γ where C is a constant independent of N and n, and the linear combination of Φ j (r, θ) is given by k,γ Σ N (ρ P H ; ρ, θ) = n d H j Φ j (ρ, θ), (8) with the coefficients dh j in (0) Proof From (4) we have ρ P H n Σ N(ρ P H n ; ρ, θ) k,γ = ρ {P H n Σ N(P H n ; ρ, θ)} k,γ This is the first result (6) Next, there exist the derivative relations, v ν = v v cos(ν, ρ) + cos(ν, θ) ρ ρθ v ρθ = v v cos(ν, θ) + cos(s, θ), ν s ρ P H n ν = ρp H cos(ν, ρ) + n ρ ρ P H n C P H n Σ N(P H n ; r, θ) k,γ (9) where ν and s are the normal and tangent directions of Γ, respectively Let v = ρ P H n, we have from (30) and (3) cos(ν, ρ) + ρ cos(ν, θ) = ρp H n cos(ν, ρ) + ρ ρ P H n Hence we obtain from (4) ν {ρ P H Σ n N(ρ P H n ; ρ, θ)} k,γ cos(ν, ρ) + ρ ν P H n ρθ P H n This gives the second result (7), and completes the proof of Lemma We have the following theorem cos (ν, θ) + ρ s P H n (30) (3) cos(s, θ) cos(ν, θ) (3) max ρ {P H Σ n N(P H ; ρ, θ)} n k,γ Γ + ν {P H Σ n N(P H n ; ρ, θ)} + P H Σ n N(P H ; ρ, θ) n k+,γ k,γ C P H Σ n N(P H ; ρ, θ) n k+,γ + ν {P H Σ n N(P H n ; ρ, θ)} (33) k,γ Theorem Let (8), (4) and (3) hold Then for w =, there exists the bound, N n N n P A Σ n N(P A ; ρ, θ) rmax n B C, (34) r max

6 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) where C is a constant independent of N and n, the special linear combination is given by and dh j ū A = Σ N N(P A ; ρ, θ) = n ( dh j Φ j (ρ, θ) + dh φ j j(ρ, θ)), (35) and dh j are given in (9) and (0), respectively Proof First we have from (4), P A n Σ N(P A n ; ρ, θ) B ρ P H n Σ N(ρ P H n ; ρ, θ) B + P h n Σ N(P h n ; ρ, θ) B, (36) and from (6) ρ P H Σ n N(ρ P H ; ρ, θ) n B C ρ P H n Σ N(ρ P H ; ρ, θ) n 0,Γ + N ν {ρ P H n Σ N(ρ P H n ; ρ, θ)} 0,Γ (37) From (8) and (3) we have P A n u, and P A n B C u B = O() Hence from (4) and (4), there exist the bounds P H n 0,Γ, P h n 0,Γ = O() (38) From (36) (38) and Lemmas and, we have n N n ρ P H (ρ, θ) Σ n N(ρ P H ; ρ, θ) rmax n B C P H (ρ, θ) n 0,Γ 0 and then from (36) and (38) C r max P A Σ n N(P A ; ρ, θ) n B ρ P H Σ n N(ρ P H ; ρ, θ) n B + P h Σ n N(P h ; ρ, θ) n B n N N rmax C { P H (ρ, θ) n B + P h (ρ, θ) n B} C r max r max r max n N n rmax, (39) n N N rmax (40) This completes the proof of Theorem Theorem Let (8), (4) and (3) hold, and choose N such that r max n N n rmax = n p Then when w =, there exists the bound, N u u A N B C N p where C is a constant independent of N and n, (4) (4) Proof From (3), () and Theorem, when w = we obtain the following error bound, N u u A N B C n N n rmax r max (43) Under (4), Eq (43) leads to u u A N B C n p (44)

7 4356 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) For (4), we may choose N n + n r max + p ln n Cn, (45) ln which implies n N We have from (44) u u A N B C N p This completes the proof of Theorem (46) Note that Eq (46) is similar to the bounds for Laplace s equation in [8] Moreover, the H errors in S may also be derived, to give the optimal convergence rate: u u N,S = O, (47) N p provided that u H p (S) (p 3) When the solution is highly smooth, the exponential convergence rates can also be obtained For the biharmonic equations with the mixed type of boundary conditions (4), the error bounds of the MAFS can also be derived, to give the polynomial convergence rates as in (46) and (47) emark The error estimates of the MFS for (8) are more challenging and difficult, details appear in a subsequent paper, although the same error bounds as (46) and (47) are obtained 3 Stability analysis on circular domains for the MAFS with Almansi s FS We consider only the circular domains in this paper For the non-circular domains, the stability analysis may follow [9] From (8), we have u N = u N (ρ, θ) = c j (ρ ln r j ) + d j ln r j = c j Φ A (ρ, θ) + j d jφ j (ρ, θ), (3) where c j and d j are the coefficients, r j = PQ j, P = ρe iθ, Q j = e iϕ j, ϕ j = π N j, and i = We also choose the uniform collocation nodes at P k = ρe iθ k and θ k = π k Then we have the N collocation equations: N u N (ρ, θ k ) = c j Φ A (ρ, θ j k) + w ρ u N(ρ, θ k ) = w c j d j φ j (ρ, θ k ) = f (ρ, θ k ), (3) ρ ΦA(ρ, θ j k) + w d j ρ φ j(ρ, θ k ) = wg(ρ, θ k ), (33) where k =,,, N, and w is a weight constant with w = /N Hence the N coefficients c j and d j can be obtained by (3) and (33) if the system of equations is nonsingular, which will be confirmed in Lemma 33 given below Denote (3) and (33) as the form of matrix and vector: Ax = b, (34) where the vectors x = {c,, c N, d,, d N } T, b = {f,, f N, wg,, wg N } T, and the matrix A N N is decomposed as [ ] A (Φ) A A = (φ), wa (DΦ) wa (Dφ) (35) (36) (37)

8 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) where Φ A A (Φ) N N (θ ) Φ A (θ N ) =, Φ A (θ N) Φ A (θ N N) φ A A (φ) N N (θ ) φ A (θ N ) =, φ A (θ N) φ A (θ N N) ρ ΦA(θ ) ρ ΦA(θ N ) A (DΦ) N N = ρ ΦA(θ, N) ρ ΦA(θ N N) ρ φa(θ ) ρ φa(θ N ) A (Dφ) N N = ρ φa(θ N) ρ φa N (θ N) (38) The matrices A (φ) and A (Dφ) result from the Dirichlet and Neumann problems of Laplace s equations on circular domains, given in [9], respectively All four sub-matrices A (Φ), A (φ), A (DΦ) and A (Dφ) are circulant Denote the eigen-matrix (see [0], p3) F ( C N N ) = N ω ω ω ω ω 4 ω (), (39) ω ω () ω ()() where ω = e π N i = cos π N + i sin π N, F is unitary with FF = F F = I, and I is the identity matrix Based on Davis [0], p 73, we have A (Φ) = F Λ (Φ)F, A (φ) = F Λ (φ)f, (30) A (DΦ) = F Λ (DΦ)F, A (Dφ) = F Λ (Dφ)F In (30), the matrices, Λ (Φ), Λ (φ), Λ (DΦ) and Λ (Dφ), are diagonal We have from [9] λ 0 (φ) O Λ (φ) =, (3) O λ (φ) λ 0 (Dφ) O Λ (Dφ) =, (3) O λ (Dφ) λ 0 (Φ) O Λ (Φ) =, O λ (Φ) λ 0 (DΦ) O Λ (DΦ) = (33) O λ (DΦ) Since the eigenvalues of similar matrices are the same, the eigenvalues of A are just those of Λ, denoted by [ ] [ ] Λ (Φ) Λ Λ = (φ) diag(λi (Φ)) diag(λ = i (φ)) (34) wλ (DΦ) wλ (Dφ) wdiag(λ i (DΦ)) wdiag(λ i (Dφ))

9 4358 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) By using matrix F in (39), we obtain from (30) [ ] A (Φ) A A = (φ) wa (DΦ) wa (Dφ) [ ] [ ] [ ] F O Λ (Φ) Λ = (φ) F O O F (35) wλ (DΦ) wλ (Dφ) O F By a permutation transformation P, λ 0 (Φ) λ 0 (φ) wλ 0 (D(φ)) wλ 0 (Dφ) P T ΛP = We have a lemma Lemma 3 Let the eigenvalues of matrix φ be λ i (Φ) λ i (φ) wλ i (DΦ) wλ i (Dφ) λ N (Φ) λ N (φ) wλ N (DΦ) wλ N (Dφ) λ k (φ) = c 0 + c ρ k + c ρ k, where c 0, c and c are constants independent of ρ Then there exist the eigenvalues for matrices Dφ, and D, (36) λ k (Dφ) = c kρ k + c (N k)ρ k, (37) λ k () = c 0 ρ + c ρ k+ + c ρ k+, (38) λ k (D) = c 0 ρ + c (k + )ρ k+ + c (N k + )ρ k+ (39) Proof Consider the matrix eigenvalue problem, Bx = λx where B is the circulant matrix dependent of ρ, and the eigenvectors are given in (39) Since ρ (B), ρ B and ρ (ρ B) are also circulant, with the same eigenvectors, the conclusions (37) (39) hold Lemma 3 For the matrix A n n with n = N from MAFS, when w =, and ρ <, there exist the leading N eigenvalues (30) λ + 0 N, λ 0, (3) Proof Since the leading eigenvalue is given from [9] by ρ N λ 0 (φ) = N ln + ε N ln, ε, we have λ 0 (Dφ) 0, λ 0 (Φ) = λ 0 (ρ φ) Nρ ln, λ 0 (DΦ) Nρ ln Hence we obtain the matrix of the leading eigenvalue λ ± 0, [ ] [ ] λ0 (Φ) λ 0 (φ) Nρ ln N ln wλ 0 (DΦ) wλ 0 (Dφ) wρn ln 0 [ ] [ ] Nρ = ln N ln Nρ = N ln (3) ρ ln 0 ρ 0 From (3) the leading eigenvalues are given by λ ± 0 to give λ Nρ λ ρn = 0, λ ± 0 = ln {Nρ ± N ρ 4 + 8ρN} = µ± ln, where µ ± satisfy

10 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) When and N is large, we have λ + 0 Nρ, λ 0 ρ, and the desired result (3) follow Lemma 33 Let all the conditions in Lemma 3 hold Then there exists the minimal eigenvalue min λ k (A) ρ N k N (33) Proof The original eigenvalues are given in [9] λ k (φ) N [ ρ k ρ ] k +, k N k k =,,, N (34) Based on Lemma 3 we have λ k (Dφ) N [ ρ k ρ ] k +, ρ [ λ k (Φ) = ρ λ k (φ) Nρ ρ k + k N k λ k (DΦ) = ρ λ k(φ) Nρ [ k + ρ k N k + + k N k Without loss of generality, let N be even Let k = N we obtain ρ N λ N (φ) ρ λ N (Φ) ρ, λ N (Dφ) N ρ N Then we obtain the matrix at k = N and w =, N [ ] ρ λk (Φ) λ k (φ) ρ wλ k (DΦ) wλ k (Dφ) w(n + 4)ρ (35) ρ k ], (36) ρ k ] (37) ρ N, (38) ρ N, λ N (DΦ) (N + 4)ρ (39) ρ N = N ρ N ρ N w N ρ N ρ ρ + 4 (330) ρ N ρ The eigenvalues of (330) are given by λ ± N λ ρ + λ 8ρ ρ N = 0 = N ρ µ ±, where µ ± satisfy the quadratic equation, (33) Similarly we have µ +, µ N (33) Then the minimal eigenvalue is obtained by min k λ k (A) = λ N (A) N ρ N This is the desired result (33) From Lemmas 3 and 33 we have the following theorem (333)

11 4360 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) Theorem 3 Let all the conditions in Lemma 3 hold Then there exists the bound max λ k (A) N k Cond = min λ k (A) CN ρ k (334) Theorem 3 is for the circular domains For the bounded simply connected domains, the exponential bounds of condition number can also be derived by following the arguments in [9] 4 Plate bending problem with smooth solutions Consider () (3) in the rectangular domain S = {(x, y) < x <, < y < }, and choose the true solution, u(x, y) = exp(x) cos y + (x + y ) exp(y) cos x (4) The plate bending problem with the mixed type of the clamped and simply supported boundary conditions is given in (4) For the collocation Eqs (7) (9), the number m of collocation nodes is often chosen to be larger than the number n of source nodes Then we obtain the over-determined system of linear algebraic equations Ax = b, where the matrix A m n (m n), x n and b m The traditional condition number is defined by Cond = σ max σ min, where σ max and σ min are the maximal and the minimal singular value of the matrix A, respectively The new effective condition number is defined in [,] as Cond_eff = b σ min x, where x is the Euclidean norm The boundary errors are given by ϵ B = u u N B = I(u N ), where I(v) = (v f ) + w (v ν g) + w (v νν g ), Γ Γ Γ with w i = /N i When Γ =, the mixed type is just the purely clamped boundary conditions We use both the traditional MFS with (8) and the MAFS with Almansi s FS with () For the clamped boundary conditions, the errors and condition numbers are listed in Tables and, where M denotes the number of uniform collocation nodes along each edge of S Then n = N and m = 8M in (4) We also use the Trefftz method with the particular solutions (4), to give the method of particular solutions (MPS), and their results are listed in Table 3 Interestingly, the numerical solutions of the MAFS are slightly better than those of the MFS However, the MPS is superior to both the MFS and the MAFS Next, we still choose the true solution (4), but use the mixed type of the clamped and simply supported boundary conditions: u = f, u ν = g, on x = ±, u = f, u νν = g, on y = ±, where ν is the exterior normal of S The errors and condition numbers of the MFS, the MAFS and the MPS are listed in Tables 4 6 All tables and figures are carried out by Java programs with double precision The errors of the MFS and the MAFS are larger than those of the MPS, because their huge Cond and Cond_eff adversely affect the accuracy of numerical solutions From the above tables and figures, we may also conclude that for smooth solutions, the errors of the MFS may catch up with those of MPS, if the huge effective condition numbers will not deteriorate the accuracy in the sense that there exist the sufficient significant digits for the numerical solutions obtained Evidently, it is due to better stability that the MPS is superior to the MFS and the MAFS 5 Plate bending problem with crack singularity 5 The MFS using uniform source points on the enlarged circle Consider the homogeneous biharmonic equation u = 0 in S, where the solution domain is also a rectangle: S : {(x, y) < x <, 0 < y < } We choose a crack model of singularity from [6], shown in Fig The section OD represents an interior crack under the clamped conditions: u = u ν = 0 The symmetric conditions, u ν = u ννν = 0 on OA BC CD, are (4) (43) (44) (45)

12 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) Table Errors and condition numbers for the biharmonic equation with the clamped boundary condition by the MFS and the MAFS with = 0, where ϵ = u u N or ϵ = u u A N MFS MAFS with Almansi s FS N M ϵ B ϵ 0,S ϵ,s Cond Cond_eff x 5 5( ) 877( ) 608( ) 54(4) ( 3) 446( 4) 689( 3) 8(6) 595(3) ( 5) 433( 6) 907( 5) 40(7) 89(5) ( 7) 37( 8) 94( 6) 3(9) 64(6) ( ) 78( ) 46( ) 3(3) 6() ( 4) 98( 5) 53( 3) 405(4) 649(3) ( 6) 806( 7) 47( 5) 785(5) 6(5) ( 7) 85( 9) 44( 7) 79(7) 89(6) Table Errors and condition numbers for the biharmonic equation with the clamped boundary condition by the MFS and the MAFS with = 50, where ϵ = u u N or ϵ = u u A N MFS NAFS with Almansi s FS N M ϵ B ϵ 0,S ϵ,s Cond Cond_eff x 5 86( ) ( ) 873( ) (7) 69() 33(3) 0 46( 6) 535( 7) 589( 6) 64() 668(5) 34(3) ( ) 7( ) 848( ) 44(4) 79(9) 59(3) 5 3( ) 606( 3) 48( ) 69(5) 654() 8() 0 44( 7) 0( 7) 09( 6) 998(8) 79(6) 5() ( ) 830( 3) 97( ) 67() 98(9) 953 Table 3 Errors and condition numbers for the biharmonic equation with the clamped boundary condition by the MPS, where ϵ = u u N N = M ϵ B ϵ 0,S ϵ,s Cond Cond_eff x 5 84( 3) 73( 3) 44( ) ( 7) 36( 7) 46( 6) 300() ( ) 55( 3) 683( ) 793() Table 4 Errors and condition numbers for the biharmonic equation with the mixed type of the clamped and simply supported boundary conditions by the MFS and the MAFS with = 0, where ϵ = u u N or ϵ = u u A N MFS MAFS with Almansi s FS N M ϵ B ϵ 0,S ϵ,s Cond Cond_eff x 5 87( ) 7( ) 07 83(4) 37() ( 3) 05( 3) 83( 3) 46(6) 648(3) ( 5) 633( 6) 08( 4) 487(7) 4(5) ( 7) 57( 8) 3( 6) 5(9) 70(6) ( ) 304( ) 7( ) 38(3) 0() ( 4) 65( 4) 74( 3) 4(4) 658(3) ( 6) 04( 4) 0( 5) 835(5) 34(5) ( 7) 9( 8) 489( 7) 85(7) 99(6) Table 5 Errors and condition numbers for the biharmonic equation with the mixed type of the clamped and simply supported boundary conditions by the MFS and the MAFS with = 50, where ϵ = u u N or ϵ = u u A N MFS MAFS with Almansi s FS N M ϵ B ϵ 0,S ϵ,s Cond Cond_eff x 5 48( ) 9( ) 89( ) 335(7) 59() 8(3) 0 8( 6) 739( 7) 693( 6) 00() 84(5) 34(3) ( ) 359( ) 8( 0) 47(4) 9(9) 59(3) 5 97( 3) 07( ) 797( ) 30(5) 8() 3() 0 346( 7) 303( 7) 46( 6) 04(9) 86(6) 5() ( ) 04( ) 0( ) 79() 39(9) 953 Table 6 Errors and condition numbers for the biharmonic equation with the mixed type of the clamped and simply supported boundary conditions by the MPS, where ϵ = u u N n = M ϵ B u u n 0,S u u n,s Cond Cond_eff x 5 04( ) 334( 3) 05( ) ( 7) 5( 7) 7( ) 870() ( ) 606( 3) 7( ) 870()

13 436 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) Fig Model II called in [6] required, where ν is the outward normal direction to the boundary S On AB when the simply supported conditions are provided, we choose the biharmonic equation with the following boundary condition, called Model II in [6] and also in this paper, see Fig u OD = 0, u y OD = 0, u y OA = 0, u yyy OA = 0, (5) u AB =, u xx AB = 0, u y BC = 0, u yyy BC = 0, u x CD = 0, u xxx CD = 0 There exists a singularity at O due to the intersection of the clamped and simply supported boundary conditions The true function can be found as in [6,] where u = (d i g i (r, θ) + c i f i (r, θ)), i= g i (r, θ) = r i+/ cos i 3 (5) θ i 3 cos i + θ, (53) i + f i (r, θ) = r i+ {cos(i )θ cos(i + )θ} (54) To compute the errors of MFS, we choose the highly accurate solution u 30 = 30 i= (d ig i (r, θ) + c i f i (r, θ)) as the exact solution u, where the coefficients d i and c i are also given in [6] For the boundary condition (5), define an energy on the boundary by I(v) = ((v ) + w v ) + xx (w v + y w 3 v ) yyy AB + CD where the weight w i (w v x + w 3 v xxx ) + = O N i BC OA (w v y + w 3 v yyy ) + DO (v + w v y ), (55), and N is the number of fundamental solution terms We use midpoint method to approximate (55) In computation, we use only the MFS with (8), and choose = 6 and = 0 In this subsection, the source points are uniformly located on the outside circle l = {(r, θ) r =, 0 θ π}, where > r max and r max = 5 Since the FS in (6) and (7) are smooth on S, the local refinements of collocation nodes P i near O should be used (see [3]) The errors and condition numbers are listed in Tables 7 and 8 From Fig, it is shown that the derivatives of these numerical solutions are large and undesirable Then we may improve the MFS by greedy adaptive techniques in the next subsection, to select the source points differently 5 Greedy adaptive techniques to select source points Greedy adaptive techniques are the algorithms to select better source points, which were first introduced for radial basis functions in [4], and then for the MFS in [5,6] The idea is as follows First we assume many potential source points, as shown in Fig 3 Then we select the effective source points, based on smaller errors For the potential source points as shown in Fig 3, we obtain Fx = b, where F = m n Then we reduce n as small as possible, to pick up the useful columns of the

14 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) Table 7 Errors and condition numbers for Model II with = 6 and M = 55 by the MFS, where ε = u u N, N is the number of FS terms, and M is the number of collocation points of AB N ε B 84( ) 359( ) 7( ) 7( ) 73( 3) ε,ab DO 79( ) 799( ) 537( ) 349( ) 50( ) ε ν,ad BC CD 50( ) 978( ) 64( ) 490( ) 44( ) ε νν,ab 00( ) 338( ) 64( ) 58( ) 645( ) ε ννν,oa BC CD 75( ) 967( ) ε 0,S 3( ) 59( ) 446( ) 648( ) 706( ) ε,s 435( ) 0( ) 77( ) 90( ) 88( ) Cond 3() 03(5) 5(7) 64(9) () Cond_eff () 85() (3) 366(3) x 687( ) 40() 437(3) 535(4) 9(6) Table 8 Errors and condition numbers for Model II with = 0 and M = 55 by the MFS, where ε = u u N, N is the number of FS terms, and M is the number of collocation points of AB N ε B ( ) 356( ) 0( ) 8( ) ε,ab DO 84( ) 83( ) 538( ) 350( ) ε ν,ad BC CD 00( ) 95( ) 84( ) 559( ) ε νν,ab 4( ) 37( ) 957( ) 578( ) ε ννν,oa BC CD 50( ) 886( ) ε 0,S 3( ) 685( ) 383( ) 639( ) ε,s 448( ) 45( ) 67( ) 87( ) Cond 05(3) 653(3) (8) 36(0) Cond_eff 3() 573() 87() 730() x 04 63() 47(4) 6(6) 4 0 Exact u x Numerical u x 05 Exact u y Numerical u y 0 u x u y x x Fig Numerical derivatives u x y=0 and u y y=0 by the MFS (N = 80 and = 6) without greedy adaptive techniques, where the source points are uniformly located on the outside circle l m n matrix F in a data-dependent way without cutting the number m of rows down for Fx = b To maintain stability, we use orthogonal transformations, but choose the columns dependent on the right-hand side vector b Let F = (f,, f n ), where f i are the column vectors To approximate b by multiples of a nonzero vector f, the error vector is given by b f bt f We may find j such that the error f T f b (bt f j ), j n, f j is minimal, where b is the Euclidean norm This is equivalent to choose the maximum of the values: (b T f j ) f j, f j 0

15 4364 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) y S x Fig 3 The initial source points Φ j φ j 5 y 05 S Fig 4 The 00(=N + N ) source points selected by Algorithm I So we have Algorithm I given in Appendix (also see [4 6]), and apply it to our problem Note that the local refinements of collocation nodes near the singular point O can also be applied Since the final source points are different for the Φ j and φ j after the greedy adaptive techniques, the FS solutions in (8) are modified as N N v N = c j Φ j (r, θ) + d j φ j (r, θ) The initial and final source points are drawn in Figs 3 and 4, respectively From Fig 4, we can see that there exist more source points selected near to the singular point O; such results coincide with [5,6] The numerical results are listed in Table 9 with N = (N + N ), and the error curves are drawn with Figs 5 and 6 From Table 9, we can see the following asymptotic relations, ε B O(090 N ), ε 0,S O(096 N ), ε,s O(096 N ), Cond O(58 N ), Cond_eff O(08 N ), where ε = u u N, and the effective condition number is given in (44) From Tables 8 and 9 at N = 40, we cite the errors: ε 0,S = 639( ), ε,s = 87( ), in Table 8, ε 0,S = 0( ), ε,s = 36( ), in Table 9, and the condition numbers: Cond = 36(0), Cond_eff = 730(), in Table 8, Cond = 5(8), Cond_eff = 90(3), in Table 9 Hence, the errors of solutions and derivatives by the MFS using the greedy adaptive techniques are smaller, while the condition numbers are smaller but the effective condition numbers are larger Comparing Figs 5 and 6 with Fig in x (56)

16 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) Table 9 Errors and condition numbers for Model II by the MFS with greedy adaptive techniques, where ε = u u N, N = (N + N ), N is the number of FS terms, and M = 55 is number of collocation points on AB N ε B 60( ) 83( 3) 6( 3) 649( 4) ( 4) 9( 4) ε 0,S 3( ) 356( ) 60( ) 0( ) 637( 3) 456( 3) ε,s 433( ) 30( ) 547( ) 36( ) 36( ) 68( ) Cond 684(3) 08(4) 646(6) 5(8) 56(0) 445(3) Cond_eff 93 39() (3) 90(3) 553(3) 0(4) x () 35(3) 66(5) 30(8) 4 Exact u x Numerical u x 08 u x x Fig 5 Numerical derivatives u x y=0 and u y y=0, by the MFS (N + N = 80) with greedy adaptive techniques 4 Exact u x Numerical u x 08 u x x Fig 6 Numerical derivatives u x y=0 and u y y=0, by the MFS (N + N = 00) with greedy adaptive techniques Section 5, the derivative curves are significantly closer to the true curves Although the greedy adaptive techniques need more CPU time, the source points selected by the greedy adaptive techniques are advantageous for the MFS for the biharmonic equation with singularity Finally in Table 0, we cite the results of the MPS from [] From Table 0, we can see that ε B = O(059 N ), Cond = O(55 N ), Cond_eff = O(09 N )

17 4366 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) Table 0 The computed results by the MPS from [], where ε = u u N, N is the number of PS terms, and σ max and σ min are the maximal and minimal singular values of the discrete matrix F, respectively N M ε B Cond Cond_eff σ max σ min ( 3) ( ) ( 5) 678(3) ( 3) ( 6) 06(5) ( 3) ( 7) (6) 86(3) 69 63( 4) ( 9) 96(6) 355(3) 34(3) 37( 4) ( 0) 958(7) 604(3) 84(4) 9( 4) ( ) 860(8) 946(3) 05(5) 3( 4) Comparing Table 0 with Tables 7 9, the solutions of the MPS are significantly more accurate than those of the MFS even using the greedy adaptive techniques Such conclusions can be seen from the data cited from Tables 9 and 0: ε B = 649( 4), Cond = 5(8), Cond_eff = 90(3), in Table 9 at N = 40, ε B = 5( ), Cond = 860(8), Cond_eff = 946(3), in Table 0 at N = 35 emark 5 Better choice of source points is an important issue for the MFS In [7 9], for Laplace s equation on the disk domain S ρ = {(r, θ) r ρ, 0 θ π}, the source points Q j are uniformly located on the larger circle l ( > ρ), and the exponential convergence rates of the MFS are derived for smooth solutions In [0], a better choice of source points is explored for the bounded simply connected domain S as follows By a conformal mapping T, the S can be transformed to a disk domain S ρ, and the source points Q j are obtained by the inverse conforming mapping: Q j = T Q j Such techniques have been implemented numerically for the MFS for the singularity problems: the biharmonic equation with the boundary conditions (5) The computed results do not show a better behavior than the MFS using the uniform source points directly on l ; details are omitted Hence, the techniques in [0] for choosing source points may not be helpful for the MFS for singularity problems Based on the computed results in this subsection, the greedy adaptive techniques are recommended emark 5 To conclude this section, let us discuss the relations among MFS, MAFS and MPS We classify them as the Trefftz method described in (9) but with different admissible functions in V N, such as FS, with Almansi s FS and PS Numerical experiments and comparisons are reported in [] for smooth solutions, to show that the MFS and MAFS achieve the equivalent convergence rates by the MPS, but offer much larger condition numbers Such comparisons retain the same for singularity problems shown in this paper These results coincide with the analysis in Sections and 3, where the optimal convergence rates and the exponential growth rates of Cond are derived In [,3], a similarity (but not an intrinsic equivalence) of algorithms between MFS and MPS is discovered for Laplace and biharmonic equations The fundamental solutions can be expanded into particular solutions (such as harmonic or biharmonic polynomials; see [4]), and a particular solution can also be expressed as a linear combination of FS; see Section However, the bounds of their errors and condition numbers are distinct; see the analysis in Sections and 3 In [5,6], the series expansions of FS are fully used for the null field method (NFM) for circular domains with circular holes, where the semi-analytic solutions are obtained Their explicit discrete algorithms and strict analysis will be reported in another paper 6 Concluding remarks To conclude this paper, let us make a few concluding remarks In this paper, the method of fundamental solution (MFS) is used for biharmonic equations with both smooth and singular solutions Both the traditional FS as r j ln r j and Almansi s FS as ρ ln r j are chosen for the MFS, the latter is denoted as the MAFS The error analysis for the MAFS is made in Section, to give the polynomial convergence rates, and in Section 3, the stability analysis of the MAFS is also made for circular domains, to give the exponential growth rates of condition number Their numerical results in Section 4 are slightly better than those by the MFS Hence, we may simply choose the MAFS with () for engineering applications It is noteworthy to pointing out that this paper is the first time to explore the analysis of the MAFS and to compare the MAFS with the traditional MFS 3 For the biharmonic equation with smooth solutions, the clamped and the mixed boundary conditions are considered The MFS, the MAFS and the MPS (ie, the Trefftz method using the particular solutions) are used, and the source points are uniformly located on the enlarged circle l with > r max The errors of the MFS and the MAFS can cope with those of the MPS, if the instability will not adversely affect the accuracy However, in computation with double precision in Section 4, the MPS is superior to both MFS and MAFS, due to better stability 4 For the biharmonic equations with crack singularity, the MPS results are given in [6] We use the MFS with source points on circles with local refinements of collocation nodes, and the computed results are not satisfactory; see Fig In general, adding one or two particular solutions is useful Whenever we find the particular solutions of the problem, we should choose the MPS However, for singularity problems, if the singular particular solutions cannot be found, we may adapt

18 Z-C Li et al / Journal of Computational and Applied Mathematics 35 (0) the MFS by the local refined collocation nodes, and by selecting the source points from the greedy adaptive techniques From the numerical results in Section 5, the greedy adaptive techniques may provide better solutions for singularity problems Acknowledgments We express our thanks to JT Chen for valuable comments and suggestions We are also indebted to TC Wu for the computation in this paper Appendix Algorithm I Step: Pick the column of F whose multiples approximate b best: Copy F A Find j such that (bt a j ) is maximal and a j 0 a j 3 Set u = a j, and v = u/ u 4 Store j Step: Transform the problem to the space orthogonal to the column: 5 A = A v(v T )A 6 b = b v(v T )b If the satisfactory solution is obtained, the computation is terminated; otherwise repeat Steps and eferences [] ZC Li, TT Lu, HY Hu, AH-D Cheng, Trefftz and Collocation Methods, WIT, Southsampton, Boston, 008 [] G Fairweather, A Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv Comput Math 9 (998) [3] A Karageorghis, G Fairweather, The Almansi method of fundamental solutions for solving biharmonic problems, Internat J Numer Methods Engrg 6 (988) [4] A Karageorghis, G Fairweather, The simple layer potential method of fundamental solutions for certain biharmonic problems, Int J Numer Methods Fluids 9 (989) 34 [5] A Karageorghis, G Fairweather, The method of fundamental solutions for the numerical solution of biharmonic equation, J Comput Phys 69 (998) [6] ZC Li, TT Lu, HY Hu, The collocation Trefftz method for biharmonic equations with singularities, Eng Anal Bound Elem 8 (004) [7] A Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM J Numer Anal (985) [8] ZC Li, Method of fundamental solutions for annular shaped domains, J Comput Appl Math 8 (009) [9] ZC Li, HT Huang, J Huang, Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace s equations, Computing 88 (00) 9 [0] P Davis, Circulant Matrices, John Wiley & Sons, New York, 979 [] ZC Li, CS Chien, HT Huang, Effective condition number for finite difference method, J Comput Appl Math 98 (007) [] ZC Li, TT Lu, Y Wei, Effective condition number of Trefftz methods for biharmonic equations with crack singularities, Numer Linear Algebra 6 (009) 45 7 [3] ZC Li, The fundamental solutions for Laplace s equation with mixed boundary condition, in: CS Chen, A Karageorghis, YS Smyrlis (Eds), The Method of Fundamental Solutions A Meshless Method, Dynamic Publishers, Inc, USA, 009, pp 9 49 Chapter [4] Schaback, H Wenland, Adaptive greedy techniques for approximate solution on large BF systems, Numer Algorithms 4 (000) [5] L Ling, Opfer, Schaback, esults on meshless collocation techniques, Eng Anal Bound Elem 30 (006) [6] Schaback, Adaptive numerical solution of MFS systems, a plenary talk at the first Inter Workshop on the Method of Fundamental Solutions (MFS007), Ayia Napa, Cyprus, June 3, 007, in: CS Chen, A Karageorghis, YS Smyrlis (Eds), The Method of Fundamental Solutions A Meshless Method, Dynamic Publishers, Inc, USA, 009, pp 8 also as Chapter [7] M Katsurada, A mathematical study of the charge simulation method II, J Fac Sci Univ Tokyo, Sect IA Math 36 (989) 35 6 [8] M Katsurada, Asymptotic error analysis of the charge simulation method in a Jordan region with an analytical boundary, J Fac Sci Univ Tokyo, Sect IA Math 37 (990) [9] M Katsurada, Charge simulation method using exterior mapping functions, Japan J Indust Appl Math (994) 47 6 [0] M Katsurada, H Okamoto, The collocation points of the method of fundamental solutions for the potential problem, Comput Math Appl 3 (996) 3 37 [] ZC Li, HJ Young, HT Huang, YP Liu, AH-D Cheng, Comparisons of fundamental solutions and particular solutions for Trefftz methods, Eng Anal Bound Elem 34 (00) [] JT Chen, CS Wu, YT Lee, KH Chen, On equivalence of Trefftz method of fundamental solutions for Laplace and biharmonic equations, Comput Math Appl 53 (007) [3] JT Chen, YT Lee, S Yu, SC Shieh, Equivalence between Trefftz method and method of fundamental solution for annular Green s function using the addition theorem and image concept, Eng Anal Bound Elem 33 (009) [4] ZC Li, MG Lee, JT Chen, New series expansions of fundamental solutions of linear elastostatics, Computing (00) (in press) [5] JT Chen, S Kuo, JH Lin, Analytic study and numerical experiments for degenerate scale problems in the boundary element method for twodimensional elasticity, Internat J Numer Methods Engrg 54 (00) [6] JT Chen, CC Hsiao, SY Leu, Null-field integral equation approach for plate problems with circular boundaries, J Appl Mech 73 (006)