Efficient Chebyshev Spectral Method for Solving Linear Elliptic PDEs Using Quasi-Inverse Technique

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1 Numer. Math. Theor. Meth. Appl. Vol. 4, No. 2, pp doi: 1.428/nmtma s.5 Ma 211 Efficient Chebshev Spectral Method for Solving Linear Elliptic PDEs Using Quasi-Inverse Technique Fei Liu, Xingde Ye and Xinghua Wang Department of Mathematics, Zhejiang Universit, Hanghou, Zhejiang 3127, China. Received 22 Januar 21; Accepted (in revised version) 16 November 21 Available online 6 April 211 Abstract. We present a sstematic and efficient Chebshev spectral method using quasiinverse technique to directl solve the second order equation with the homogeneous Robin boundar conditions and the fourth order equation with the first and second boundar conditions. The ke to the efficienc of the method is to multipl quasiinverse matri on both sides of discrete sstems, which leads to band structure sstems. We can obtain high order accurac with less computational cost. For multi-dimensional and more complicated linear elliptic PDEs, the advantage of this methodolog is obvious. Numerical results indicate that the spectral accurac is achieved and the proposed method is ver efficient for 2-D high order problems. AMS subject classifications: 65N35, 65N22, 65F5, 35J5 Ke words: Chebshev spectral method, quasi-inverse, Helmholt equation, Robin boundar conditions, general biharmonic equation. 1. Introduction Due to high order accurac, spectral methods have gained increasing popularit for several decades, especiall in the field of computational fluid dnamics (see, e.g.,[1, 2] and the references therein). According to different test functions in a variational formulation, there are three most common spectral schemes, namel, the collocation, Galerkin and tau methods. Since the collocation methods approimate differential equations in phsical space, it is ver eas to implement and adaptable to various of problems, including variable coefficient and nonlinear differential equations. Weideman and Redd constructed a MATLAB software suit to solve differential equations b the spectral collocation methods in[13]. Trefethen s book[12] eplained the essentials of spectral collocation methods with the aid of 4 short MATLAB programs. For multi-dimensional problems, the spectral collocation methods discretie the differential operators emploing Kronecker products. In Corresponding author. addresses: Ð Ù Ñ ÐºÓÑ (F. Liu), Ü Ò Ý Þ Ùº ÙºÒ (X. Ye), Ü Ò Ù Ñ ÐÝÛ Ò ºÒ Ø (X. Wang) c 211 Global-Science Press

2 198 F. Liu, X. Ye and X. Wang the Galerkin method, we work in the spectral space, it ma lead to well conditioned linear sstems with sparse matrices for problems with constant coefficients b choosing proper basis functions (see, e.g.,[3,5,9,1]). Although the collocation and Galerkin methods usuall lead to optimal error estimates, the primar drawback of collocation method is that the differentiation matrices are dense in all dimensions, and it is generall not feasible to solve multi-dimensional problems b emploing the Galerkin method. Shen used a matri diagonaliation method to solve the 2-D and 3-D Helmholt problems in[9] and[1], but an eigenvalue-eigenvector decomposition of the discretied linear operator is required. Therefore it can onl be used for relativel simple differential equations. Heinrichs[6] utilied a Galerkin basis set to obtain efficient differentiation matrices, and eploited the inherent structure of both the Galerkin differentiation matrices and the relationship between the Chebshev and Galerkin spectral coefficients to maimie the sparsit of differential operators. Julien and Watson[7] presented the quasi-inverse technique to efficientl solve linear elliptic differential equations with constant coefficients under Dirichlet boundar conditions. In this paper, we present an etension of the Chebshev spectral method using quasi-inverse technique to directl solve the Helmholt equation with the homogeneous Robin boundar conditions and the general biharmonic equation with the first and second boundar conditions. For the general biharmonic equation, we give a uniform treatment for the first and second boundar conditions. The main idea is that we emplo a truncated series of Chebshev polnomials to approimate the unknown function, and the differential operator is epanded b Chebshev polnomials which vector of coefficients is represented b the product of derivative matri and vector of Chebshev coefficients of unknown function. The coefficients of this series are taken to be equal to the coefficients of the right-hand side epansion. According to Galerkin basis satisfing boundar conditions, we identif a transformation matri which transforms the Chebshev and Galerkin coefficients, and then multipl a quasi-inverse matri on both sides of the resulting spectral sstem to obtain a pre-multiplied sstem A v=b f, where A and B have band structure. After we solve this sstem of equations, the Galerkin spectral coefficients are converted back to Chebshev spectral coefficients. We obtain the approimation solution from spectral space to phsical space using the forward Chebshev transform b FFT. The remainder of the paper is organied as follows. In the net section, we introduce some notations and summarie a few mathematical facts used in the remainder of the paper. In Section 3, we consider the Helmholt equations for one, two and three dimensional cases. In Section 4, we stud the general biharmonic equations for one and two dimensional cases. In Section 5, we present some numerical results. Finall, some concluding remarks are given in Section Notation 2. Preliminaries We introduce some basic notations as follows:

3 Chebshev Spectral Method for Solving Linear PDEs 199 δ mn, Kronecker-delta function, equal to 1 if m=n and ero otherwise, Kronecker matri product T m (), mth degree Chebshev polnomial ū, vector of spectral coefficients associated with Chebshev spectral modes v, vector of spectral coefficients associated with Galerkin spectral modes I, identit matri with respect to I (±), quasi-identit matri with respect to. It is an identit matri with rows of eros at the top/bottom for±, respectivel D p, pth differential matri with respect to D p, pth quasi-inverse matri for D p E (±), "shifted identit" matri with ones on the ± super/sub-diagonal for the variable S {v}, transformation matri for the unknown v in the spatial direction. It is used to transform between Chebshev spectral coefficients and Galerkin spectral coefficients 2.2. Chebshev polnomials The Chebshev polnomials can be represented b trigonometric functions which satisf the orthogonalit relation T m ()=cos mθ, θ= arccos, (T m (), T n ()) ω = c mπ 2 δ mn, (2.1) where the weight functionω()=(1 2 ) 1 2, and c = 2 and c m = 1 for m 1. The derivative of the Chebshev polnomials can be represented b T m According to the trigonometric identit we can obtain the following relation msin(mθ) ()=. sinθ 2 sinθ cos mθ= sin(m+1)θ sin(m 1)θ, 2T m ()= T m+1 () m+1 T m 1 (), m>1. (2.2) m 1 The Chebshev epansion of a function u L 2 ω ( 1,1) is u()= û m T m (), û m = 2 1 u()t m ()ω()d. (2.3) πc m 1 m=

4 2 F. Liu, X. Ye and X. Wang The derivative of u epanded in Chebshev polnomials can be represented formall as where u ()= û (1) m = 2 c m m= p=m+1 p+m odd û (1) m T m(), (2.4) pû p, m. (2.5) This epression is a consequence of the relation (2.2). From (2.2) one has 2mû m = c m 1 û (1) m 1 û(1) m+1, m 1, (2.6) and from (2.6), we have the following recursion relation c m û (1) m = 2(m+ 1)û m+1+ û (1) m+2, m, (2.7) which ields (2.5). The generaliation of this relation is[2] c m û (q) m = 2(m+1)û(q 1) m+1 + û(q) m+2, m. (2.8) Similarl, the second derivative of u is epanded b Chebshev polnomials u ()= m= Due to the recursion relation (2.8), the coefficients are û (2) m = 1 c m p=m+2 p+m even û (2) m T m(). (2.9) p(p 2 m 2 )û p, m. (2.1) Here, we can define a differential matri D 2 such that ū(2) = D 2 ū, where D2 is an upper triangular matri with eros on the main diagonal, ū (2) and ū are the vectors of û (2) m and û m, respectivel Quasi-inverse matri Definition 2.1. D p is called the quasi-inverse matri of order p associated with D p in the spatial direction, if D p D p = I (p) and D p D p = I ( p). The following properties of the quasi-inverse matri can be found in[7]. Propert 1. A necessar condition of the definition of the quasi-inverse is that the matri D p has eros in the first p rows and the last p columns.

5 Chebshev Spectral Method for Solving Linear PDEs 21 Propert 2. Structure from the basis polnomials translates to the quasi-inverse representation such that there is a well defined structure between different order operators, (1) D 2 = D D, (2) D p = I (p) D p I ( p), (3) D p D q I (p) D p+q. The quasi-inverse matri D 1 associated with D 1 is a tri-diagonal matri whose nonero entries defined b the three term recursion relation (2.6) are: ḋ m,m 1 = c m 1 2m, sub-diagonal ḋ m,m+1 = e 1 m M. (2.11) m+2 2m, super-diagonal The quasi-inverse matri D 2 associated with D 2 is a penta-diagonal matri whose nonero elements are[4]: c m 2 ḋ m,m 2 = 4m(m 1), 2nd sub-diagonal ḋ m,m = e m+2, main diagonal 2 m M, (2.12) ḋ m,m+2 = 2(m 2 1) e m+4 4m(m+1), 2nd super-diagonal where e m = 1 for m M, e m = for m> M. The non-ero entries of D p analticall b the three term recursion relation are defined 2.4. Kronecker products c m û (q) m û(q) m+2 = 2(m+1)û(q 1) m+1, m. (2.13) Definition 2.2. If A and B are of dimensions p q and r s, respectivel, then the Kronecker product A B is the matri of dimension pr qs with p q block form, where the i, j block is a i j B. Propert 3. If A C and B D eist, then(a B) (C D)=(A C) (B D). In multiple dimensions, we can separate the differential operator b emploing the Kronecker product. For instance, consider the discretiation of the 2nd derivative operator in 1-D, 2-D and 3-D shown below: 1D u () D 2 ū 2D u (, ) (D 2 I ) ū 2D u (, ) (D 1 D1 ) ū 2D u (, ) (I D 2 ) ū 3D u (,,) (D 2 I I ) ū 3D u (,,) (D 1 D1 I ) ū, etc.

6 22 F. Liu, X. Ye and X. Wang 3. Helmholt equations In this section, we are interested in emploing the Chebshev spectral method using quasi-inverse technique to solve the Helmholt equation where I=( 1,1) and d= 1,2,3. αu u= f, inω= I d, (3.1) D Helmholt Let us begin with the one dimensional Helmholt equation with the homogeneous Robin boundar condition αu() u ()= f(), I, (3.2) a ± u(±1)+ b ± u (±1)=. (3.3) We first epand u(), u () and f() in terms of Chebshev polnomials respectivel where u M ()= M û m T m (), ū=(û,û 1,,û M ) T, (3.4) m= u M ()= M f M ()= m= û (2) m T m(), ū (2) =(û (2),û(2) 1,,û(2) M )T, (3.5) M ˆf m T m (), f=(ˆf,ˆf 1,,ˆf M ) T, (3.6) m= û (2) m = 1 c m M p=m+2 p+m even p(p 2 m 2 )û p. (3.7) In spectral-galerkin method, it is essential to seek an appropriate basis functions to satisf the boundar condition. We usuall choose a compact combination of Chebshev polnomials as basis function. The following lemma provides a basis function which satisfies the homogeneous Robin boundar condition[11]. Lemma 3.1. Let us define a m = {(a + + b + (m+2) 2 )( a + b m 2 ) (a b (m+2) 2 )( a + b + m 2 ) /DET m, b m = (a + + b + (m+1) 2 )( a + b m 2 ) +(a b (m+1) 2 )( a + b + m 2 ) /DET m, (3.8)

7 Chebshev Spectral Method for Solving Linear PDEs 23 with DET m = 2a + a +(m+1) 2 (m+2) 2 (a b + a + b 2b b + ). (3.9) If DET m, then a linear combination of Chebshev polnomials φ m ()= T m ()+ a m T m+1 ()+ b m T m+2 () (3.1) satisfies the homogeneous Robin boundar condition(3.3). Proof. Since T m (±1)=(±1) m and T m (±1)=(±1)m 1 m 2, the boundar condition (3.3) leads to the following sstem for{a m, b m } : (a + + b + (m+1) 2 )a m +(a + + b + (m+2) 2 )b m = a + b + m 2, (a b (m+1) 2 )a m +(a b (m+2) 2 )b m = a + b m 2, (3.11) whose determinant DET m is given b (3.9). When DET m,{a m, b m } can be uniquel determined from (3.11). Remark 3.1. We note in particular that If a ± = 1 and b ± = (Dirichlet boundar condition), we have a m =, b m = 1. Hence,φ m ()= T m () T m+2 () satisfies the homogeneous Dirichlet boundar condition. If a ± = and b ± = 1 (Neumann boundar condition), we have Hence, a m =, φ m ()= T m () b m = m2 (m+2) 2. m 2 (m+2) 2 T m+2() satisfies the homogeneous Neumann boundar condition. We choose{φ m } M 2 m= as Galerkin basis function and represent u() in terms of a truncated series of Galerkin basis function: u M ()= M 2 m= ũ m φ m (), v=(ũ,ũ 1,,ũ M 2 ) T. (3.12) Since the Galerkin basis functionφ m () are linear combinations of Chebshev polnomials, the Chebshev coefficients û m are also linear combinations of the coefficients ũ m, we tr to seek a transformation matri S, between the Chebshev and Galerkin spectral representation such that ū=s v. (3.13) Here v should be added two fictitious modes ũ M 1 and ũ M, which we specif to be identicall ero, since S is(m+ 1) (M+ 1).

8 24 F. Liu, X. Ye and X. Wang To determine the appropriate transformation matri S i, we first project onto each T n () mode b appling inner products to each side defined b (2.1), (φ m (), T n ()) ω =(T m ()+ a m T m+1 ()+ b m T m+2 (), T n ()) ω = c mπ 2 δ c m+1 π c m+2 π mn+a m δ m+1,n + b m δ m+2,n. (3.14) 2 2 We can epress this inner product relation in terms of shifted identit matrices E (k) [e m,n ] [δ m,n k ]. For k<, E (k) defines a square matri(m+ 1) (M+ 1) with ones on the kth sub-diagonal, whereas for k>, the identit matri I =[δ m,n ] is equivalent to E (). A matri with entries[e m,n δ m,n+1 ] can be represented b E ( 1). We define the following transformation matri for the Robin boundar condition, S = E () + E ( 1) W 1 + E ( 2) W 2, (3.15) where W 1 and W 2 are the weight matrices with diagonal entries{a m } and{b m }, respectivel. The transformation matri provides a convenient means for discretiing the differential equation with the boundar conditions embedded within the differentiation matri. The discrete 1-D Helmholt sstem of equations in spectral space is (αi D 2 ) S v=i f. (3.16) To increase the efficienc, the original sstem (3.16) can be replaced b A v=b f, where A and B have band structure. We eploit the quasi-inverse technique, and multipl both sides b D 2 (αd 2 I (2) ) S v=d 2 f. (3.17) We obtain the 1-D pre-multiplied Galerkin operator A (αd 2 I (2) ) S (left) and quasiinverse B D 2 (right) which are banded matrices. Since ũ M 1 and ũ M are identicall Restricted Matri A Restricted Matri B n = n = 39 ÙÖ ½ ÆÓÒ¹Þ ÖÓ ÒÞµ Ð Ñ ÒØ Ó 1¹ ÔÖ ¹ÑÙÐØ ÔÐ Ö ØÖ Ø Ð Ö Ò ÓÔ Ö ØÓÖ A Рص Ò ÕÙ ¹ ÒÚ Ö B Ö Øµ ÖÓÑ Õº º½ µ Û Ø 17 Ý Ú Ö º

9 Chebshev Spectral Method for Solving Linear PDEs 25 ero, we can solve the(m 1) (M 1) sub-sstem which is called "restricted sstem", where we ignore the top two rows and the last two columns of matrices A and B. The non-ero elements of A and B take the form Fig. 1. The cost to solve sstem of equations is (M) operations. After we solve this sstem of equations, v should be converted back to Chebshev spectral coefficients ū via the relation (3.13), which requires (M) operations since S has the special structure. We obtain the approimation solution of u() from spectral space to phsical space using the forward Chebshev transform b FFT. Remark 3.2. We note that for other boundar conditions, we onl need identif the corresponding Galerkin basis function and don t need to derive a new spectral Chebshev differential matri for each Galerkin basis set b using the transformation matri. The method for solving elliptic differential equations can easil generalie to two and three dimensional cases D Helmholt Consider the 2-D Helmholt equation αu u= f, inω= I 2, with the homogeneous Robin boundar conditions Let us denote (a ± u+ b ± u )(±1, )=, (c ± u+ d ± u )(,±1)=. (3.18) u(, )= M 2,N 2 m,n= ũ m,n φ m ()φ n (), whereφ m () andφ n () are similar to that in one dimension case. The transformation matrices for each Galerkin basis are the same as the are in one dimension S = E () + E ( 1) W 1 + E ( 2) W 2 and similarl for S. The discrete 2-D Helmholt sstem of equations in spectral space is α(i I ) (D 2 I + I D 2 ) (S S ) v=(i I ) f, (3.19) where v and f are vectors of length(m 1) (N 1) formed b the columns of matrices(ũ m,n ) M 2,N 2 m,n= and( f m,n ) M 2,N 2 m,n=, respectivel. We eploit the quasi-inverse technique similarl to the 1-D case, and multipl both sides b D 2 α(d 2 ) (I(2) + D 2 I (2) ) (S S ) v=(d 2 ) f, and emplo the Propert 3 to obtain α(d 2 S S ) (I (2) S ) (D 2 S ) (D 2 S ) (I (2) S ) v =(D 2 ) f. (3.2)

10 26 F. Liu, X. Ye and X. Wang Restricted Matri A Restricted Matri B n = n = 576 ÙÖ ¾ ÆÓÒ¹Þ ÖÓ ÒÞµ Ð Ñ ÒØ Ó 2¹ ÔÖ ¹ÑÙÐØ ÔÐ Ö ØÖ Ø Ð Ö Ò ÓÔ Ö ØÓÖ A Рص Ò ÕÙ ¹ ÒÚ Ö B Ö Øµ ÓÖ Ø 2¹ À ÐÑ ÓÐØÞ ÔÖÓ Ð Ñ Û Ø Ý Ú Ö º From Fig. 2, we can see the strict band structure of 2-D pre-multiplied Galerkin operator A α(d 2 S S ) (I (2) S ) (D 2 S ) (D 2 S ) (I (2) S ), and quasi-inverse B D 2 D 2. There are(m 1) (N 1) unknowns and the bandwidth grows as (M+N), so the cost to solve sstem of equations is (MN 2 + M 2 N) operations D Helmholt The formulation of the 3-D Helmholt equation with the homogeneous Robin boundar conditions is similar to the 2-D case. We utilie the same Galerkin basis functions and transformation matrices as in the 1-D and 2-D problems. The discrete 3-D Helmholt sstem of equations for (3.1) in spectral space is α(i I I ) (D 2 I I + I D 2 I + I I D 2 ) (S S S ) v =(I I I ) f. We eploit the quasi-inverse technique similarl to the 1-D case, and multipl both sides b D 2 α(d 2 ) I (2) D 2 I (2) (S S S ) v=(d 2 ) f. The original sstem is replaced b A v= B f, where A=α(D 2 S S S ) (I (2) (D 2 S ) (I (2) S ) (D 2 D 2 I (2) S ) (D 2 S ) (D 2 S ) S ) (D 2 S ) (D 2 S ) (I (2) S ),

11 Chebshev Spectral Method for Solving Linear PDEs 27 and B=D 2 which again has a well defined band structure., Remark 3.3. Since the Dirichlet and Neumann boundar conditions are the special cases of the Robin boundar condition, we can solve the Helmholt equation with the common three boundar conditions for 1-D, 2-D and 3-D. For nonhomogeneous boundar problems, we alwas first reduce them to problems with homogeneous boundar conditions b modifing the right-hand side, then solve the homogeneous boundar problems. Algorithm 3.1. The Chebshev spectral method for PDEs using quasi-inverse technique involves the following steps: 1. Identif the appropriate Galerkin basis functions to fulfill the boundar conditions; 2. For each Galerkin basis in each coordinate, identif the appropriate transformation matri S i ; 3. Evaluate the Chebshev coefficients{ˆf m } M m= of I M f() from{f( j )} M j= (backward Chebshev transform) and evaluate f ; 4. Discretie the differential equation using the standard Kronecker formalism to obtain (D) S i v=i i f ; 5. Identif the quasi-inverse matri B for the highest order operator in each spatial direction; 6. Multipl the sstem on both sides b B to obtain a pre-multiplied sstem A v=b f ; 7. Solve the sstem for v; 8. Convert from Galerkin basis v to Chebshev basis ū; M 9. Evaluate u M ( j )= û m T m ( j ), j=,1,, M (forward Chebshev transform). m= 4. General biharmonic equations In this section, we generalie the methodolog of quasi-inverses to high order problems. Consider the general biharmonic equation with the following two tpes of boundar conditions 2 u α u+βu= f, inω= I d, (4.1) u Ω = u n =, (4.2) Ω

12 28 F. Liu, X. Ye and X. Wang and u Ω = 2 u n 2 =, (4.3) Ω where n is the normal vector to Ω and d= 1,2. (4.2) and (4.3) are named the first and second boundar conditions respectivel. Here, we give a uniform treatment to solve the first-kind general biharmonic problem constituted b (4.1) and (4.2) and the second-kind general biharmonic problem constituted b (4.1) and (4.3) D General biharmonic equations Consider the 1-D general biharmonic equation u (4) () αu ()+βu()= f(), I, (4.4) with the first boundar conditions u(±1)=u (±1)=and the second boundar conditions u(±1)=u (±1)=, respectivel. Here, we would like to seek the basis functions of the form φ m ()= T m ()+ a m T m+2 ()+ b m T m+4 (). Lemma 4.1. For the first boundar condition(4.2), we define a m = 2(m+2) m+3, b m= m+1 m+3. For the second boundar condition(4.3), we define 2(m+2)(15+ 2m(m+4)) a m = (m+3)(19+ 2m(6+ m)), b m= (m+1)(3+2m(m+2)) (m+3)(19+ 2m(6+ m)). Then φ m ()= T m ()+ a m T m+2 ()+ b m T m+4 () (4.5) satisfies the first and second boundar conditions respectivel. Proof. Since T m (±1)=(±1) m and T m (±1)= 1 3 (±1)m m 2 (m 2 1), the boundar condition u(±1)= leads to the following equation for{a m, b m } a m + b m = 1. (4.6) The boundar condition u (±1)= leads to the following equation (m 4 + 8m m m+12)a m +(m m m m+ 24)b m = (m 4 m 2 ). (4.7) B solving the above sstem (4.6) and (4.7), we obtain{a m, b m } for the second boundar conditions (4.3). Similarl for the first boundar conditions.

13 Chebshev Spectral Method for Solving Linear PDEs 29 We identif the transformation matri for this Galerkin basis (4.5) S = E () + E ( 2) W 2 + E ( 4) W 4, (4.8) where W 2 and W 4 are the weight matrices with diagonal entries{a m } and{b m }, respectivel. We can now write down the matri form of the discretied equation (D 4 αd2 +β I ) S v=i f, (4.9) where v has been padded with four eros at the bottom of the column. We eploit the quasi-inverse technique similarl to the 1-D Helmholt equation, and multipl both sides b D 4 (I (4) αd 4 D 2 B emploing the Propert 2, we obtain I (4) (I αd 2 +β D 4 ) S v=d 4 f. +β D 4 I ( 4) ) S v=d 4 f. (4.1) The top four rows on both sides are identicall ero, so it is trivial to identif appropriate sub-sstem, shown in Fig. 3. The banded sstem can be solved in (M) operations. The conversion from the Galerkin basis to Chebshev basis is still (M), so the total cost is quasi-optimal. Restricted Matri A Restricted Matri B n = n = 25 ÙÖ ÆÓÒ¹Þ ÖÓ ÒÞµ Ð Ñ ÒØ Ó 1¹ ÔÖ ¹ÑÙÐØ ÔÐ Ö ØÖ Ø Ð Ö Ò ÓÔ Ö ØÓÖ A Рص Ò ÕÙ ¹ ÒÚ Ö B Ö Øµ ÖÓÑ Õº º½¼µ Û Ø 49 Ý Ú Ö º D General biharmonic equations The power of this methodolog is its read etensibilit to higher dimensions and more complicated sets of differential equations. Consider the 2-D general biharmonic equation (4.1) with the first and second boundar conditions respectivel. We discretie u(, ) with the product of two Galerkin basis functions u(, )= M 4,N 4 m,n= ũ m,n φ m ()φ n (),

14 21 F. Liu, X. Ye and X. Wang whereφ m () is defined as in the 1-D case and similarl forφ n (). The transformation matrices for each Galerkin basis are the same as the are in one dimension. The discrete 2-D general biharmonic sstem of equations in spectral space is D 4 I + 2(D 2 D2 )+ I D 4 α(d2 I + I D 2 )+β(i I ) (S S ) v=(i I ) f. (4.11) We eploit the quasi-inverse technique similarl to the 2-D Helmholt equation again, and multipl both sides b D 4 D 4, obtain A v= B f, where A=I (4) S D 4 S + 2(I (4) + D 4 S I (4) α(i (4) D 2 D 2 S I (4) D 2 S ) S +β(d 4 S D 4 S ) S D 4 S + D 4 S I (4) D 2 S ), (4.12) and B=D 4 D 4. Because of the structure, it is eas to identif the trivial equations and etract the restricted sstem, see Fig. 4. Restricted Matri A Restricted Matri B n = n = 36 ÙÖ ÆÓÒ¹Þ ÖÓ ÒÞµ Ð Ñ ÒØ Ó 2¹ ÔÖ ¹ÑÙÐØ ÔÐ Ö ØÖ Ø Ð Ö Ò ÓÔ Ö ØÓÖ A Рص Ò ÕÙ ¹ ÒÚ Ö B Ö Øµ ÓÖ Ø 2¹ Ò Ö Ð ÖÑÓÒ ÔÖÓ Ð Ñ Û Ø 2 2 Ý Ú Ö º Remark 4.1. Because the number of unknowns is the same, the compleit of the solve is roughl the same as that of the 2-D Helmholt equation, although the bandwidth is slightl wider, it still onl grows like M+ N. Remark 4.2. In the case of the second-kind general biharmonic problem constituted b (4.1) and (4.3), one can split the governing equation (4.1) into a set of two weakl coupled Poisson equations which can be efficientl solved b emploing the proposed method. Although we onl stud the standard second and fourth order problems, general linear problems can be similarl solved b quasi-inverse methodolog.

15 Chebshev Spectral Method for Solving Linear PDEs Numerical results In this section, we give some numerical results obtained b using the algorithms presented in the previous sections. All test codes are implemented in MATLAB and are performed on desktop Dell PC with single core processor. We utilie MATLABs built-in "sparse" representation for matrices and solve the sstem of equations via the "backslash" operator. The timing is performed b averaging the time to solve each test problem several separate runs for each number of unknowns. Ì Ð ½ ÇÒ ¹ Ñ Ò ÓÒ À ÐÑ ÓÐØÞ ÕÙ Ø ÓÒ Û Ø Ø Ö ØÝÔ ÓÙÒ ÖÝ ÓÒ Ø ÓÒ º Dirichlet BC Neumann BC Robin BC M Error Cond(A) Error Cond(A) Error Cond(A) Cond(B) E E E E E E E E E E E E E E E E E E E+4 Eample 5.1. Consider the following one dimensional Helmholt equation αu() u ()= f(), I. Given the eact solution under three tpes boundar conditions. The homogeneous Dirichlet boundar condition and the eact smooth solution u(±1) =, u( ) = sin(π). (5.1) The homogeneous Neumann boundar condition and the eact smooth solution The homogeneous Robin boundar condition u (±1)=, u()=cos(π). (5.2) a ± u(±1)+ b ± u (±1)=, (5.3) where a ± =π and b ± = 1. Given the following known function the eact solution can be verified to be f=(π 2 +α)(sin(π)+cos(π)), u()=sin(π)+cos(π). Table 1 lists the maimum pointwise error of u u M, condition numbers of matrices A and B using the quasi-inverse technique withα=1under the three tpes boundar conditions. Numerical results of this eample show that Chebshev spectral method via quasiinverse technique converges eponentiall under the three tpes boundar conditions, because the difference onl lies in the different transformation matrices corresponding the Galerkin basis functions. Compared to the condition number of matri A with Dirichlet boundar condition (Dirichlet BC), those of matri A with Neumann and Robin BCs are larger.

16 212 F. Liu, X. Ye and X. Wang Eample 5.2. Consider the following two dimensional Helmholt equation αu u= f, inω= I 2. The homogeneous Dirichlet boundar condition and an eact smooth solution are u Ω =, u(, )=sin(π) sin(π). (5.4) The homogeneous Neumann boundar condition and an eact smooth solution are u n Ω=, u(, )=cos(π) cos(π). (5.5) Ì Ð ¾ ÌÛÓ¹ Ñ Ò ÓÒ À ÐÑ ÓÐØÞ ÕÙ Ø ÓÒ Û Ø ØÛÓ ØÝÔ ÓÙÒ ÖÝ ÓÒ Ø ÓÒ º Dirichlet BC Neumann BC M, N Error Cond(A) CPU(s) Error Cond(A) CPU(s) Cond(B) E-3 6.2E E E-4 1.3E E E E E E-2 1.4E-7 8.E E E E E E E E E E E E+6 3.7E E E E E E E E E E E E+8 In Table 2, we list the maimum pointwise error of u u MN, condition numbers of matrices A and B and time of solving numerical solutions using the proposed Chebshev spectral method with α = 1 under Dirichlet and Neumann boundar conditions. The results indicate that the spectral accurac is achieved and the Chebshev spectral method using quasi-inverse technique is ver efficient even for 2-D problems. There is no an difference for the Dirichlet and Neumann boundar conditions, it shows the methodolog can be adopted to deal with more complicated boundar problems. Eample 5.3. Consider the 1-D general biharmonic equation with the first boundar conditions: u (4) () αu ()+βu()= f(), I, u(±1)=u (±1)=, (5.6) with an eact smooth solution u()=sin 2 (4π). This eample was used in[1] and[11]. In Table 3, we list the maimum pointwise error of u u M and time of solving sstem of equations A v=b f with two tpical choices ofα,β. The results indicate that the spectral accurac is achieved for bothαand β are eros and non-eros cases. Because the matrices A and B have band structure, we can efficientl solve the sstem of equations A v=b f, and reduce the computing time. Eample 5.4. Consider the 2-D first-kind general biharmonic boundar-value problem (4.1) and (4.2) with an eact smooth solution u(, )=[1+cos(π)][1+cos(π)]/π 4.

17 Chebshev Spectral Method for Solving Linear PDEs 213 Ì Ð Å Ü ÑÙÑ ÔÓ ÒØÛ ÖÖÓÖ Ò Ø Ñ Û Ø M+ 1 Ý Ú Ö º M α β Error CPU(s) α β Error CPU(s) E E-5 2M 2 M E E E E-5 2M 2 M E E E E-5 2M 2 M E E-4 This problem was solved withα=,β= via a spectral collocation method based on integrated Chebshev polnomials in[8]. For the sake of comparison, we measure the accurac of a numerical solution via the norm of relative errors of the solution N e (u)= M,N (u (e) i,j u i,j ) 2 i,j=, M,N (u (e) i,j )2 i,j= where u (e) i,j and u i,j are the eact and computed values of the solution u at point(i, j). In Table 4, we list the relative error N e (u) and time of solving sstem of equations A v= B f with two tpical choices ofα,β. Although the results obtained are slightl less accurate than the proposed integration formulation (PIF) in[8], we can efficientl solve the sstem for 2-D fourth-order problems because of band structure. We draw the mesh plot of solution of the general biharmonic equation (4.1) withα=,β =, which is indistinguishable from the eact solution in Fig. 5. Finall, we compare the actual CPU cost for solving a 2-D Poisson equation and a 2- D biharmonic equation via the proposed algorithms in the previous sections. The two- Mesh of biharmonic equation U ÙÖ ÆÙÑ Ö Ð ÓÐÙØ ÓÒ Ó Ø 2¹ ÖÑÓÒ ÔÖÓ Ð Ñ Û Ø Ý Ú Ö º

18 214 F. Liu, X. Ye and X. Wang Ì Ð Ê Ð Ø Ú ÖÖÓÖ N e (u) Ò Ø Ñ ÓÖ Å Æ Û Ø (M+ 1) (N+ 1) Ý Ú Ö µº M+ 1 α β N e (u) CPU(s) α β N e (u) CPU(s) E E-4 (M+ 1) 2 (N+ 1) E E E E-3 (M+ 1) 2 (N+ 1) E E E E-3 (M+ 1) 2 (N+ 1) E E E E-3 (M+ 1) 2 (N+ 1) E E-3 dimensional Poisson equation is u=32π 2 sin(4π) sin(4π), inω= I 2, u Ω =, (5.7) with an eact smooth solution u(, ) = sin(4π) sin(4π ). The two-dimensional biharmonic equation with the first boundar conditions: 2 u= f inω= I 2, u Ω = u n Ω=, (5.8) with an eact smooth solution u(, )=(sin(2π) sin(2π)) 2. The two equations were solved b the Legendre-Galerkin method in[9]. In Table 5, we list in the second column the maimum pointwise error of u u MN and time of solving approimate solution in parentheses of Poisson equation (5.7); in the third column, we list the maimum pointwise error of u u MN and time of solving approimate solution in parentheses of biharmonic equation (5.8). It is obvious that the approimate solutions converge eponentiall to the eact solution. Comparing with the actual time in[9] (the eecution time plus the preprocessing time) via Legendre-Galerkin method, the CPU time in seconds via the proposed algorithms is less. It is worth noting that the actual time for solving a 2-D biharmonic equation is 2.5 times of that for solving a 2-D Poisson equation. Ì Ð Error Ò ÈÍ Ø Ñ Ó Ø ÈÓ ÓÒ Ò ÖÑÓÒ ÓÐÚ Ö º M, N Poisson Error (CPU) biharmonic Error (CPU) E-6 (.24) E-5 (.53) E-11 (.37) E-11 (.99) E-15 (.65) E-14 (.175) E-15 (.91) E-14 (.264) 6. Conclusion We have presented a sstematic Chebshev spectral method using quasi-inverse technique to efficientl solve linear elliptic PDEs. B multipling the quasi-inverse matri on the sstem s both sides, we obtain the sstem of equations which has band structure, so it can be efficientl solved. We can achieve the same numerical accurac compared with other methods with less computational cost. The advantages of this methodolog are eas to solve the multi-dimensional and more complicated linear elliptic PDEs with a few common boundar conditions. We note that the Chebshev spectral method via quasi-inverse

19 Chebshev Spectral Method for Solving Linear PDEs 215 technique to solve the 2-D general biharmonic equations is ver competitive to other eisting numerical methods. Acknowledgments The authors would like to thank Professor Jie Shen and the anonmous referees for their valuable comments and suggestions on this work. This work was partiall supported b the grants of National Natural Science Foundation of China (No , 18112) and Chinese Universities Scientific Fund No. 21QNA319. References [1] C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, T. A. ZANG, Spectral Methods in Fluid Dnamics, Springer-Verlag, New York, [2] C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, T. A. ZANG, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, New York, 26. [3] E. H. DOHA AND A. H. BHRAWY, Efficient spectral-galerkin algorithms for direct solution for second-order differential equations using Jacobi polnomials, Numer. Algorithms., 42 (26), pp [4] H. DANG-VU AND C. DELCARTE, An accurate solution of the Poisson equation b the Chebshev collocation method, J. Comput. Phs., 14 (1993), pp [5] E. H. DOHA AND W. M. ABD-ELHAMEED, Efficient spectral-galerkin algorithms for direct solution for second-order equations using ultraspherical polnomials, SIAM J. Sci. Comput., 24: 2 (22), pp [6] W. HEINRICHS, Improved Condition Number for Spectral Methods, Math. Comput, 53: 187 (1989), pp [7] K. JULIEN AND M. WATSON, Efficient multi-dimensional solution of PDEs using Chebshev spectral methods, J. Comput. Phs., 228 (29), pp [8] N. MAI-DUY AND R. I. TANNER, A spectral collocation method based on integrated Chebshev polnomials for two-dimensional biharmonic boundar-value problems, J. Comput. Appl. Math., 21 (27), pp [9] J. SHEN, Efficient spectral-galerkin method I. Direct solvers for the second and fourth order equations using Legendre polnomials, SIAM J. Sci. Comput., 15: 6 (1994), pp [1] J. SHEN, Efficient spectral-galerkin method II. Direct solvers of second and fourth order equations b using Chebshev polnomials, SIAM J. Sci. Comput., 16: 1 (1995), pp [11] J. SHEN AND T. TANG, Spectral and High-Order Methods with Applocations, Science Press, Beijing, 26. [12] L. N. TREFETHEN, Spectral Methods in MATLAB, PA: SIAM, Philadelphia, 2. [13] J. A. C WEIDEMAN AND S. C REDDY, A MATLAB differentiation matri suite, ACM Trans. Math. Softw., 26: 4 (2), pp