FINITE ELEMENT DUAL SINGULAR FUNCTION METHODS FOR HELMHOLTZ AND HEAT EQUATIONS

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1 J. KSIAM Vol.22, No.2, , 2018 ttp://dx.doi.org/ /jksiam FINITE ELEMENT DUAL SINGULAR FUNCTION METHODS FOR HELMHOLTZ AND HEAT EQUATIONS DEOK-KYU JANG AND JAE-HONG PYO DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY, CHUNCHEON 24341, KOREA address: ABSTRACT. Te dual singular function metod(dsfm is a numerical algoritm to get optimal solution including corner singularities for Poisson and Helmoltz equations. In tis paper, we apply DSFM to solve eat equation wic is a time dependent problem. Since te DSFM for eat equation is based on DSFM for Helmoltz equation, it also need to use Serman- Morrison formula. Tis formula requires linear solver n + 1 times for elliptic problems on a domain including n reentrant corners. However, te DSFM for eat equation needs to pay only linear solver once per eac time iteration to standard numerical metod and perform optimal numerical accuracy for corner singularity problems. Because te Serman-Morrison formula is rater complicated to apply computation, we introduce a simplified formula by reanalyzing te Serman-Morrison metod. 1. INTRODUCTION In order to deal wit corner singularity solution, let te computational domain Ω be an open and bounded concave polygon in R 2. We assume tat Ω as one reentrant corner for simple explanation. It can be readily applied on multiple reentrant corner domain and is performed in 4. Te goal of tis paper is to construct te dual singular function metod to solve eat equation: u t µ u = f in Ω, (1.1 u = g on Ω, wit given initial value u(t = 0, x = (x in Ω and termal diffusivity µ > 0. Altoug te given function f is very smoot, te solution of (1.1 ave singular beavior near te reentrant corner. Tis corner singularity make lose accuracy of numerical solution trougout te wole domain. To overcome tis difficulty, tere are several primary ways. One is locally adaptive mes refinement or moving mes tecnique. Oter ways are so called Singular Function Metod (SFM (see, e.g., [9] and Dual Singular Function Metod (DSFM constructed in Received by te editors Marc ; Accepted June ; Publised online June Matematics Subject Classification. 65N12, 65M12, 65M60, 35A20. Key words and prases. Dual singular function metod, corner singularity, eat equation. Corresponding autor. 101

2 102 D.-K. JANG AND J.-H. PYO [1, 2, 7, 8]. DSFM for Poisson equation as been introduced in [3, 4] and ten it is extended to solve Helmoltz equations in [10]: µ u + ku = f in Ω, u = g on Ω. Because it is known in [5] tat te singular function of eat equation (1.1 can be written by a linear combination of tat of Helmoltz equation (1.2, we can construct DSFM to solve eat equation (1.1 by combining a time discrete sceme and results in [10]. DSFM as to use Serman-Morrison formula wic requests linear solver 2 times for any elliptic problem, but DSFM for eat equation does not need linear solvers per time iteration cost to standard numerical algoritm. Because te Serman-Morrison formula is rater complicated to apply real computation, we introduce a simplified version by reanalyzing te Serman-Morrison formula. Tis paper organized as follows. In section 2, we summarize te results of DSFM for Helmoltz equation in [10], and we establis te relationsip between regular part of te solution and discrete singular functions by analyzing Serman-Morrison formulation (see Teorem 2.4. We will construct DSFM to solve eat equation (1.1 in section 3 and present several numerical results in section DUAL SINGULAR FUNCTION METHOD TO SOLVE HELMHOLTZ EQUATION (1.2 We denote Ω = Γ in Γ out, were Γ in is part of boundary including reentrant corner and Γ out = Ω Γ in. Let ω be te internal angle of Γ in satisfying π < ω < 2π. Te singular and te dual singular functions of Poisson equation are summarized in [3, 4] as were s P (r, θ := r π/ω Θ(θ and s P d (r, θ := r π/ω Θ(θ, (2.1 Θ(θ = c cos(αθ + d sin(αθ. (2.2 We note tat c and d will be determined to old te boundary conditions of u near te corner on Γ in. We introduce te following lemma in [3]. Lemma 2.1. Te functions s P and s P d is armonic functions. Te singular function s H H 1 (Ω and te dual singular function s Hd L 2 (Ω of Helmoltz equation (1.2 ave to satisfy µ s H + ks H = 0 and µ s Hd + ks Hd = 0. (2.3 Te functions are written by power series s H (r, θ = r (1 π/ω + and s Hd (r, θ = r π/ω (1 + k i ν i 4 i i! i m=1 (m + αr2i k i ν i 4 i i! i m=1 (m αr2i Θ(θ (2.4 Θ(θ, (2.5

3 A FEM-DSFM FOR HELMHOLTZ AND HEAT EQUATIONS 103 were Θ(θ is defined in (2.2. We note ere s Hd / H 1 (Ω. Because r β is in H 2 (Ω if β 1, te singular parts of (2.4 and (2.5 are only S(r, θ := r π/ω Θ(θ and S d (r, θ := r π/ω Θ(θ. (2.6 So we can define singular and dual singular functions as in (2.1. We note ere tat S(r, θ and S d (r, θ are not satisfying (2.3 and it is an obstacle to construct DSFM for Helmoltz equation. We need to use a cut-off function to be 0 boundary condition, because S(r, θ and S d (r, θ of (2.6 is not 0 on Γ out. To do tis, we use notation B(r 1 ; r 2 = {(r, θ : r 1 < r < r 2 and 0 < θ < ω} Ω. We now define smoot cut-off function η ρ as 1 in B(0; 1 2 ρ, η ρ (r = p(r in B( 1 2ρ; ρ, 0 in Ω\B(0; ρ, (2.7 wit p(r is a very smoot function and ρ is a parameter determining te range of p(x. Ten te solution u of (1.2 can be rewritten by u = w + αη ρ S, (2.8 were α is called te stress intensity factor and w H 2 (Ω is regular part of solution. Te main idea of DSFM is to find α and w instead of computing u / H 2 (Ω. We start to construct DSFM wit rewriting te Helmoltz equation by inserting (2.8 into (1.2 µ w + kw + α( µ η ρ S + kη ρ S = f in Ω, w = g on Ω. (2.9 Since η ρ equals 0 identically in Γ out and S = 0 on Γ in, te function η ρ u S becomes 0 on Ω. So u and w ave same boundary values. To derive DSFM for eat equation, we test (2.9 by v H 1 0 to get w, v + kw, v + α µ η ρ S + ks, v = f, v. (2.10 Te variational formulation (2.10 as 2 unknowns (w, α, so we need one more equation and te equation is a single equation, because α is a real number. linearly independent to above system. not disappeared α. computable near te singularity corner. So we test wit te dual singular function S d (r, θ / H 1 (Ω to obtain µ w + kw, η 2ρ S d + α µ η ρ S + kη ρ S, η 2ρ S d = f, η 2ρ S d. (2.11 We ave a crucial lemma wic is proved in [10]. Lemma 2.2. Dual Singular function S d as te following properties.

4 104 D.-K. JANG AND J.-H. PYO 1. S d L 2 and S d / H ϕ, η 2ρ S d = ϕ, η 2ρ S d, for all ϕ H 1 0. In conjunction wit Lemma 2.2, (2.11 can be re written by α = 1 (β f w, µ η 2ρ S d + ks d, (2.12 were = µ η ρ S + ks, η 2ρ S d and β f = f, η 2ρ S d. DSFM is an algoritm to solve te system (2.11 and (2.12. In order to construct fully discrete algoritm of te system, we introduce finite element space V := {v H 1 0 (Ω : v K P(K K T}, were P(K is a polynomial function space degree p. In ligt of inserting α in (2.12 into (2.11, te discrete system becomes w V and α R satisfying, for all v V, w, v 1 w, µ η 2ρ S d + kη 2ρ S d µ η ρ S + kη ρ S, v Te matrix form of (2.13 is + kw, v = f, v β f µ η ρ S + kη ρ S, v. (2.13 (A + ab T w = F, (2.14 were te matrix A is te discrete linear operator of te Hemoltz equation. Also a, b T and 1 F in are generated by µ η ρ S + kη ρ S, v, w, µ η 2ρ S d + kη 2ρ S d and f, v β f µ η ρ S + kη ρ S, v, respectively. Te linear system (2.14 can be solved by te Serman- Morrison formulation in [6]: (A + ab T 1 = A 1 A 1 ab T A 1 a + b T A 1 a. (2.15 Te well posedness of (2.13 as been proved and obtain te following finite element approximation in [10]. Teorem 2.3. Let (w, α be te solution of (2.9 and w H 2 (Ω H 1 0 (Ω. And let (w, α be te solution of te system (2.12 and (2.13. Ten tere exists a positive constant 0 suc tat 0, for all, satisfying w w 1 C f 0, w w 0 C 1+α f 0, α α C 1+α f 0. Te Serman-Morrison formulation (2.15 is crucial to solve (2.14, but (2.15 requires linear solver 2 times and is rater complicated to apply to real computation, especially multiple singularities problems and time evolution problems like eat equation. So we reanalyze te Serman-Morrison formulation (2.15 wit solutions u and z of u, v + ku, v = f, v, v V, (2.16

5 and A FEM-DSFM FOR HELMHOLTZ AND HEAT EQUATIONS 105 z, v + kz, v = µ η ρ S + kη ρ S, v, v V, (2.17 receptively. Ten we arrive at te following result. Teorem 2.4. If we define ζ v = v, µ η 2ρ S d + kη 2ρ S d, for any function v V, ten w wic is te solution of (2.13 is ( βf ζ u w = u z. (2.18 ζ z Proof. In ligt of definition of ζ w, (2.13 can be rewritten by, for all v V, w, v + kw, v ζ w µ η ρ S + kη ρ S, v = f, v β f µ η ρ S + kη ρ S, v. Since b T A 1 F R, te Serman-Morrison formulation (2.15 yields From (2.16 and (2.17, w = A 1 F A 1 ab T A 1 a + b T A 1 a F = A 1 F bt A 1 F 1 + b T A 1 a A 1 a. A 1 F = u β f z. Also, A 1 a = z and b T v = ζ v /, for v V implies tat Terefore, (2.19 becomes b T A 1 a = ζ z / and b T A 1 F = ζ u w = u β f z ζ u ( βf = u te proof is complete. + β f ζ βs 2 z z 1 ζz β sζ u β f ζ z z ( ζ z ( βf ( ζ z ζ u + β f ζ z = u ( ζ z We so construct te following DSFM using (2.4. z = u + β f βs 2 ζ z. ( βf ζ u ζ z z. (2.19 Algoritm 1 (FE-DSFM for Helmoltz equation. Let w be discrete smoot part and let α be discrete stress intensity factors in (2.13.

6 106 D.-K. JANG AND J.-H. PYO Step 1: Find (u, z as te solution of (2.16 and (2.17. Step 2: Compute w by (2.18. Step 3: Compute α by second part of (2.13. We now apply Algoritm 1 to more complicated problem including several reentrant corners. Te solution u can be expressed by n u = w + α i η ρi S i, and te Helmoltz equation becomes n µ w + kw + α i ( µ η ρi S i + kη ρi S i = f in Ω, w = g on Ω. (2.20 We can coose cut off functions η ρi and η 2ρj to ave mutually disjoint support to old µ η ρi S i + ks i, η 2ρj S dj = 0 if i j. In ligt of testing (2.20 by η 2ρi S di, we readily obtain α i = 1 (β fi w, µ η 2ρi S di + ks di. (2.21 i Te finite element weak form of system (2.20 and (2.21 is to find w V and α i R, i = 1 n, satisfying, for all v V, n w, v + kw, v + α i µ η ρi S i + kη ρi S i, v = f, v, α i = 1 (2.22 (β fi w, µ η 2ρi S di + kη 2ρi S di, i and its matrix form becomes n (A + a i b T i w = F. Let ζ i,v = v, µ η 2ρi S di + ks di and z i be te solution of z i, v + kz i, v = µ η ρi S i + kη ρi S i, v, v V. (2.23 In conjunction wit Teorem 2.4, w would be n ( βfi ζ u w = u i ζ zi z i. (2.24 Finally we arrive at DSFM to solve Helmoltz equations (1.2 on domain Ω including several reentrant corners. Algoritm 2 (Multiple singularities version of Algoritm 1. Let w be discrete smoot part and let α i be discrete stress intensity factors of α i Step 1: Find u and z i, i = 1... n, as te solution of (2.16 and (2.23.

7 A FEM-DSFM FOR HELMHOLTZ AND HEAT EQUATIONS 107 Step 2: Compute w by (2.24. Step 3: Compute α by second part of ( DUAL SINGULAR FUNCTION METHOD FOR THE HEAT EQUATION In tis section, we extend Algoritm 1 to solve te eat equation (1.1 using te backward Euler time discrete formula (BDF1 and te second order backward Euler time discrete formula (BDF2. We first consider BDF1 for eat equation: u u n µ u = f. (3.1 τ We already know in [5] tat te form of singular solution is same to tat of Helmoltz equation. It means tat te solution of (1.1 can be expressed by and (3.1 can be rewritten by u(x, t = w(x, t + α(tη ρ S, (3.2 (w µτ w + α (η ρ S µτ η ρ S = τf + w n + α n η ρ S. (3.3 Since (3.3 and (2.9 ave similar expression, te same strategy to Algoritm 1 leads us te solver of (3.3: find w V and α R satisfying, for all v V, were µτ w, v + w, v + α η ρ S µτ η ρ S, v = τ f, v + w n, v + α n η ρs, v, α = 1 ( τβ β f + w n + α n η ρ S, η 2ρ S d ζ w, s = η ρ S µτ η ρ S, η 2ρ S d, β f = f(t, η 2ρ S d, (3.4 ζ w = w, η 2ρ S d µτ η 2ρ S d. Te matrix form of te system becomes (2.14 and it can be solved by (2.15. To construct algoritm for eat equations, we denote z be te solution of µτ z, v + z, v = µτ η ρ S + η ρ S, v, v V. (3.5 Ten we arrive at DSFM wit BDF1 sceme to solve eat equation. Algoritm 3 ( FE-DSFM wit backward Euler for Heat equation. Let z be te solution of (3.5 and set û 0 = u0. Repeat for 1 n N: Step 1: Find u as te solution of µτ u, v + u, v = τ f, v + û n, v, v V. Step 2: Compute w w = u ( τβf + û n, η 2ρS d ζ u z. ζ z

8 108 D.-K. JANG AND J.-H. PYO Step 3: Compute α by second part of (3.4. Step 4: Update û = w + α η ρ S. Remark 1. Te Serman-Morrison formulation (2.15 requires to solve linear system 2 times for Poisson and Helmoltz equations, but Algoritm 3 need to apply linear solver only 1 at eac iteration, so tis algoritm use same linear solvers per eac time iteration to standard metod. And Algoritm 3 as optimal accuracy beavior for corner singularities problem, in contrast standard numerical metod. We now construct DSFM for eat equation wit backward Euler formula (BDF2 time discretization: 3u 4u n + u n 1 µ u = f. 2τ By te same manner wit te case BDF1, (3.2 leads us 3w 2µτ w + α (3η ρ S 2µτ η ρ S and testing (3.6 wit v V and wit η 2ρi S di yield were = 2τf + 4(w n + α n η ρ S (w n 1 + α n 1 η ρ S, 2µτ w, v + 3 w, v + α 3η ρ S 2µτ η ρ S, v = 2τ f, v + 4( w n, v + α n η ρs, v ( w n 1, v + α n 1 η ρ S, v, v V, α = 1 (F β ζ w, s F = 2τ f, η 2ρ S d + 4( w n, η 2ρ S d + α n η ρ S, η 2ρ S d ( w n 1, η 2ρ S d + α n 1 η ρ S, η 2ρ S d, ζ w = w, 3η 2ρ S d 2µτ η 2ρ S d and βs = 3η ρ S 2µτ η ρ S, η 2ρ S d. Since tis system also constructs a linear system of te form we can find w by Teorem 2.4, tat is were z is te solution of w = u (A + ab T w = F, ( F ζ u ζ z (3.6 (3.7 z, (3.8 2µτ z, v + 3 z, v = 2µτ η ρ S + 3η ρ S, v, v V.

9 If we denote û n = wn +αn η ρs and û n 1 of, for all v V, A FEM-DSFM FOR HELMHOLTZ AND HEAT EQUATIONS 109 2µτ u, v + 3 u, v Finally, we arrive at BDF2 DSFM to solve eat equation (1.1. = w n 1 +α n 1 η ρ S, ten u becomes te solution = 2τ f, v + 4 û n, v û n 1, v. (3.9 Algoritm 4 ( FE-DSFM wit BDF2 for Heat equation. Let w be discrete smoot part and α be discrete stress intensity factors of α(t of n + 1 step. Initially given u 0, set w 1 and α 1 using te backward FE-DSFM. Repeat for 1 n N Step 1: Find u as te solution of (3.9. Step 2: Compute w by (3.8. Step 3: Compute α by second part of ( NUMERICAL TEST In tis section, we document te computational performance of eac algoritm witin a polygonal domain wit reentrant corners. We use cut-off function in (2.7 wit p(r = 15 { 8 ( 4r ρ ( 4r 3 3 ρ 3 1 ( 4r 5 } 5 ρ 3. All computations are performed wit te conforming P 1 finite element and 6 points quadrature rule. Example 4.1. We consider Helmoltz equations (1.2 on te Γ sape computational domain ([ 1, 1] [ 1, 1] ([0, 1] [ 1, 0]. We coose te smoot part of te solution as { sin(2πx(1/2y 2 + y(y 2 1, (y < 0 w = and set te exact solution sin(2πx( 1/2y 2 + y(y 2 1, (y 0, u = w + αη ρ S wit α = 3.0 and ρ = 0.4. Also, te singular function S is given by S(r, θ = r π/ω sin( π ω θ wit ω = 3 2π. Te forcing term f is determined to become k = 1.0 and ν = 1.0. Table 1 is te error decay of te standard finite element metod for Example 4.1. Te losing convergence orders is natural beavior because of u / H 2 (Ω. Table 2 is te error decay of te DSFM wit Algoritm 1 and displays optimal error decay in contrast wit results in Table 1 Te next example is a multiple corner singularities case.

10 110 D.-K. JANG AND J.-H. PYO TABLE 1. Error table for Example 4.1 wit Standard FEM. u u 0 Iu u u u 1 Errors Order Errors Order Errors Order 1/ e e e-00 1/ e e e / e e e / e e e / e e e / e e e TABLE 2. Error table for Example 4.1 wit Algoritm 1. w w 0 Iw w w w 1 α α Errors Order Errors Order Errors Order Errors Order 1/ e e e e-03 1/ e e e e / e e e e / e e e e / e e e e / e e e e Example 4.2. Let te computational domain be ([ 2, 2] [ 2, 2] ([ 1, 1] [ 1.1] including 4 reentrant corners. Let smoot part of te solution be given by { sin(πx(1/2y 2 + y(y + 1(y + 2, (y < 0 w = sin(πx( 1/2y 2 + y(y 1(y 2, (y 0. In tis experiment, we coose k = 1.0, ν = 1.0 and set te forcing term f to be te exact solution u = w + α 1 η ρ1 S 1 + α 2 η ρ2 S 2 + α 3 η ρ3 S 3 + α 4 η ρ4 S 4, were ρ 1 = ρ 2 = ρ 3 = ρ 4 = 0.4 and stress intensity factors α 1 = 1.0, α 2 = 2.0, α 3 = 3.0 and α 4 = 4.0. Table 3 is te mes analysis of Algoritm 2 wic is based on (2.24. Tis results also display optimal error decay rates. We now perform numerical simulations for te eat equations. Example 4.3. We consider eat equation wit Γ sape computational domain ([ 1, 1] [ 1, 1] ([0, 1] [ 1, 0]. We use t as time variable and we perform Algoritms 3 and

11 A FEM-DSFM FOR HELMHOLTZ AND HEAT EQUATIONS FIGURE 1. Domain, mes and η ρi S i, (i = 1 4 for Example 4.2 TABLE 3. Error table for Example 4.2 wit Algoritm 2. w w 0 Iw w w w 1 4 α i α i Errors Order Errors Order Errors Order Errors Order 1/ e e e e-02 1/ e e e e / e e e e / e e e e / e e e e / e e e e wit smoot part of solution { sin(t sin(2πx(1/2y 2 + y(y 2 1, (y < 0 w = sin(t sin(2πx( 1/2y 2 + y(y 2 1, (y 0, and exact solution u = w + exp(tη ρ S in [0, 1] Ω, were ρ = 0.4 and forcing term is also determined by ν = 1.0. In tis example, te stress intensity factor depends on variable t and Tables 4 6 are mes analysis at t = 1. Since backward Euler metod is only 1st order sceme for time, we set τ = 16 2 to ceck te optimal convergence rates and Table 4 sows te optimal results of Algoritm 3. We apply te BDF2 time discretization sceme, Algoritm 4, for te same example. Since it is second order metod, we set τ = 1 2, and so tis computational cost is muc more effective tan Algoritm 3. Table 5 is results of te standard finite element metod wit BDF2 time discretization for Example 4.3. Since u / L 2 ([0, 1] : H 2, te errors do not decay wit optimal order.

12 112 D.-K. JANG AND J.-H. PYO TABLE 4. Error table for Example 4.3 wit Algoritm 3. w w 0 Iw w w w 1 α α τ Errors Order Errors Order Errors Order Errors Order 1/8 1/ e e e e-03 1/16 1/ e e e e /32 1/ e e e e /64 1/ e e e e /128 1/ e e e e /256 1/ e e e e TABLE 5. Error table for Example 4.3 wit Standard FEM wit BDF2 time discretization. u u 0 Iu u u u 1 τ Errors Order Errors Order Errors Order 1/8 1/ e e e-00 1/16 1/ e e e /32 1/ e e e /64 1/ e e e /128 1/ e e e /256 1/ e e e TABLE 6. Error table for Example 4.3 wit Algoritm 4. w w 0 Iw w w w 1 α α τ Errors Order Errors Order Errors Order Errors Order 1/8 1/ e e e e-03 1/16 1/ e e e e /32 1/ e e e e /64 1/ e e e e /128 1/ e e e e /256 1/ e e e e Table 6 is error table for Algoritm 4 for Example 4.3 and te errors converge to 0 wit optimal rate. So we can conclude tat DSFM for eat equation as optimal convergence rate for corner singularity problems. Finally, Figure 2 is error evolutions of smoot part of u on te L 2 space and errors of stress intensity factors. It displays smoot error propagation on time.

13 A FEM-DSFM FOR HELMHOLTZ AND HEAT EQUATIONS 113 Error 1.5 x 10 3 w w wit =1/32 τ =1/64 0 α α wit =1/32 τ =1/64 w w wit =1/64 τ =1/ α α wit =1/64 τ =1/ Time FIGURE 2. Error evolution of Example 4.3 wit Algoritm 4. ACKNOWLEDGMENT Tis study was supported by 2015 Researc Grant from Kangwon National University (No REFERENCES [1] H. Blum and M. Dobrowolski, On finite element metods for elliptic equations on domains wit corners, Computing 28 (1982, [2] M. Bourlard, M. Dauge, M.-S. Lubuma, and S. Nicaise, Coeficients of te singularities for elliptic boundary value problems on domains wit conical points III. Finite element metods on polygonal domains, SIAM Numer. Anal. 29 (1992, [3] Z. Cai and S. Kim, A finite element metod using singular functions for te Poisson equation: Corner singularities, SIAM J. Numer. Anal. 39 (2001, [4] Z. Cai, S. C. Kim and B. C. Sin, Solution metods for te Poisson equation: corner singularities, SIAM J. SCI. COMPut. 23 (2001, [5] H.J. Coi and J.R. Kweon Te fourier finite element metod for te corner singularity expansion of te eat equation Computers Mat. Applic. 69 (2015, [6] Gene H. Colub and Carles F. Van Loan Matrix Computations, Te Jon Hopkins University Press (1996. [7] M. Djaoua, Equations Intégrales pour un Probleme Singulier dans le Plan, Tese de Troisieme Cycle, Universite Pierre et Marie Curie, Paris, [8] M. Dobrowolski, Numerical approximation of elliptic interface and corner problems, Habilitation-Scrift, Bonn, [9] G. J. Fix, S. Gulati and G. I. Wakoff, On te use of singular functions wit finite elements approximations, J. Comput. Py. 13 (1973, [10] S.C. Kim, J.S. Lee and J.H. Pyo A finite element metod using singular functions for Helmoltz equations: Part I, J. KSIAM 12 (2008,