u = 0 in Ω, u = f on B,

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1 Preprint April 1991 Colordo Stte University Deprtment of Mthemtics COMPUTATION OF WEAKLY AND NEARLY SINGULAR INTEGRALS OVER TRIANGLES IN R 3 EUGENE L. ALLGOWER 1,2,4, KURT GEORG 1,3,4 nd KARL KALIK 3,5 Abstrct. We study the pproximtion of wekly singulr integrls over tringles in generl position in R 3, giving explicit formule where convenient nd numericl qudrture in more generl cses. Prticulr models considered concern the colloction nd Glerkin methods in the boundry integrl pproch to the Dirichlet problem for Lplce s eqution. Key words. singulr integrls, numericl qudrture, boundry element method, symptotic estimtes AMS(MOS) subject clssifictions. 65D30, 65R20, 65N35, 45E05 1. Introduction. In the boundry integrl pproch to solving prtil differentil equtions over domin Ω R 3 one converts to n integrl eqution over B = Ω nd pproximtes the domin Ω by polytope whose boundry pproximtes B. The fces of the polytope re polygons, so there is no loss of generlity in ssuming tht consists of tringles. See, e.g., [2, sec. 15.4] for n introduction to piecewise liner methods for pproximting surfces. An efficient implementtion including smoothing steps hs been developed in [11]. To motivte our generl discussion, let us consider the problem of finding solution u to the first kind integrl eqution (1.1) u(x) x ξ d x = f(ξ), ξ, where denotes the Eucliden norm in R 3. Eqution (1.1) stems from the boundry integrl pproch for the Dirichlet problem u = 0 in Ω, u = f on B, where B is pproximted by, see, e.g., the books [3,4,5,7,8]. The numericl solution of (1.1) is usully crried out vi colloction or Glerkin method. Our discussion below pertins to both methods. Let Q 1,...,Q N be the vertices of nd let 1,..., M be the fces (tringles) of. We will ssume tht the domin Ω nd hence the pproximte boundry re bounded. 1 Prtilly supported by the Ntionl Science Foundtion under grnt number DMS Prtilly supported by the Deutscher Akdemischer Austuschdienst under study visit. 3 Prtilly supported by the Deutsche Forschungsgemeinschft. 4 Deprtment of Mthemtics, Colordo Stte University, Ft. Collins, Colordo Mthemtisches Institut A, Universität Stuttgrt, D(W)-7000 Stuttgrt, Germny. 1

2 2 e. l. llgower, k. georg nd k. klik Let us seek n pproximte solution to (1.1) which is of the form u A i ϕ i, i=1 where the ϕ i re chosen bsis functions nd the A i R re unknown coefficients which must be clculted. Typiclly, the ϕ i re piecewise polynomils. Tht is, ϕ i (x) = B i,q,α x α for x q, where we dopt the convention α n x =(x 1,x 2,x 3 ) q, Z + : = set of non-negtive integers, α =(α 1,α 2,α 3 ) Z 3 +, α :=α 1 + α 2 + α 3, x α :=x α 1 1 xα 2 2 xα 3 3. The non-negtive integer n denotes the degree of the polynomil ϕ i. In the colloction method for (1.1) the unknown coefficients A 1,...,A n re determined by solving the system of liner equtions (1.2) A i i=1 ϕ(x) x Q p d x = f(q p ), p =1,...,N. In (1.2) it is necessry to compute the entries of the coefficient mtrix: i,p := ϕ(x) x Q p d x = M q=1 α n B i,q,α q x α x Q p d x. Consequently, it becomes necessry to clculte integrls of the form x (1.3) I(p, q, α) := α x Q p d x. In the Glerkin method for (1.1) system of equtions of the form q (1.4) A i i=1 ϕ j (ξ) ϕ i (x) x ξ d x d ξ = ϕ j (ξ) f(ξ) d ξ must be stisfied for j =1,...,N. Here the ϕ i re the sme generl bsis functions s those described in the colloction method.

3 Now writing computtion of singulr integrls over tringles 3 I i (ξ) := ϕ i (x) x ξ d x, the coefficients of the system (1.4) ssume the form i,j = ϕ j (ξ) I i (ξ) d ξ. Typiclly, the coefficients i,j re computed vi qudrture formul: i,j c p ϕ(q p ) I i (Q p ) p=1 where the c p, p =1,...,N re weights. Since I i (ξ) = = q=1 ϕ i (x) x ξ d x α n B i,q,α q x α x Q p d x, we re gin led to the tsk of computing integrls of the form (1.3) lso in the Glerkin method. Our im therefore is to develop methods for evluting or pproximting integrls of the form (1.3). 2. Geometric Trnsformtions. Recent relted works concerning pproximtion of surfce integrls in the context of boundry element methods re [1,6,9,10]. The book of Hckbusch [5] discusses the pnel method in detil nd gives some cses in which certin integrls over tringles cn be integrted exctly. Since the tringles q re generlly in n rbitrry position in spce, we begin the tretment of the computtion of (1.3) by describing briefly trnsformtions which bring the generl configurtion of Q p nd q into more stndrd configurtion. Let us suppose tht Q p nd q re described in stndrd 3-dimensionl Crtesin co-ordintes (x 1,x 2,x 3 ), nd the vertices of q re denoted by v 1,v 2,v 3. An ffine trnsformtion x y =(y 1,y 2,y 3 ) brings the generl configurtion into stndrd configurtion hving the following properties: (i) q lies in the plne y 3 = 0. Tht is, y q implies y =(y 1,y 2, 0). (ii) The point Q p lies on the y 3 -xis. Tht is, Q p =(0, 0,c) for some c R. By introducing the orthonorml bsis (2.1) ρ 1 := v2 v 1 v 2 v 1,ρ3 := (v2 v 1 ) (v 3 v 1 ) (v 2 v 1 ) (v 3 v 1 ),ρ2 := ρ 3 ρ 1,

4 4 e. l. llgower, k. georg nd k. klik we obtin y = Ax b or x = A t (y + b), where A :=(ρ 1,ρ 2,ρ 3 ) t,b:= ρ1 v 1 ρ 2 Q p. ρ 3 Q p It is immeditely seen tht x α = d γ y γ 1 1 yγ 2 2 for x q, γ α where γ =(γ 1,γ 2 ) Z 2 + nd d γ R re uniquely determined nd esily computed from the bove trnsformtion. Hence we cn concentrte upon pproximting integrls of the form (2.2) Î(p, q, γ) := q γ y 1 1 yγ 2 2 y Q p d y, since (1.3) is merely liner combintion of the integrls in (2.2). Let us denote by v 0 := (0, 0, 0) the projection of Q p into the y 3 = 0 plne, nd the vertices v i of q re understood to be expessed in the y-co-ordintes. The indexing of the vertices v 1, v 2, v 3 will be chosen to correspond to the orienttion of, i.e., ρ 1 defined in (2.1) is the outer norml of on q. The tringle q oriented in this wy will be written s q =[v 1,v 2,v 3 ]. The prmeter c ppering in the y-co-ordintes of Q p =(0, 0,c) will ply n importnt rôle below. By mking use of the orienttion, it follows tht for ny integrnd F which is defined on ll the domins involved, the following formul holds: F (y) d y = F (y) d y + F (y) d y + F (y) d y. [v 1,v 2,v 3 ] In prticulr, we hve where (2.3) Î i,j (p, q, γ) := [v 0,v 1,v 2 ] [v 0,v 2,v 3 ] Î(p, q, γ) =Î1,2(p, q, γ)+î2,3(p, q, γ)+î3,1(p, q, γ) [v 0,v i,v j ] [v 0,v 3,v 1 ] y γ 1 1 yγ 2 2 y Q p d y for i, j {1, 2, 3}, i j. For convenience, let us denote ny integrl of the form (2.3) by J(γ,c), where c is the y 3 -co-ordinte of Q p nd the indices of the vertices of the tringles hve been supressed. Note tht we hve reduced our problem to the evlution of such integrls. 3. Numericl Approximtion of J(γ,c). We begin the clcultion of J(γ,c) by introducing the polr co-ordintes (3.1) y 1 = ρ cos(θ), y 2 = ρ sin(θ).

5 computtion of singulr integrls over tringles 5 Then (3.2) J(γ,c)= v i b R(θ) (θ) sin γ 2 (θ) 0 ρ γ +1 dρ dθ, ρ2 + c2 where the limits of the integrtion R(θ),, b re derived corresponding to the following figure : R(θ) d µ v j b v 0 β θ α Figure (3.3) Hence we hve π 2 = ( ) v tn 1 i 2 v1 i π 2 ( ) b = tn 1 v j 2 R(θ) = v j 1 d cos(θ β) = if v i 1 =0, if v i 1 0, if v j 1 =0, if v j 1 0, d cos(µ), β = π 2 α, µ = θ β, π 2 ( ) α = tn 1 v j 2 vi 2 v j 1 vi 1 if v j 1 = vi 1, if v j 1 vi 1.

6 6 e. l. llgower, k. georg nd k. klik We will consider the integrl J(γ,c) in the form (3.4) J(γ,c)= where b (3.5) G p (r, c) := (θ) sin γ 2 (θ) G γ +1 (R(θ),c) dθ, r Integrtion by prts yields the recursion formul G 0 (r, c) =ln ( r + r 2 + c 2) ln( c ), 0 ρ p ρ2 + c 2 dρ. (3.6) G 1 (r, c) = r 2 + c 2 c, G p (r, c) = 1 p rp 1 r 2 + c 2 p 1 p c2 G p 2 (r, c) for p =2, 3,... Note tht the function G p (r, c) cn be efficiently clculted vi this recursion. The formule hold for r>0 nd c 0, nd re supplemented by G p (r, 0) = 1 p rp for r>0 nd p>0. It is cler tht for fixed c R nd p =1, 2,..., the function r G p (r, c) isc, nd consequently the integrnd in (3.4) is C. Hence, the integrl J(γ,c) cn be pproximted by highly ccurte nd efficient numericl qudrture methods. We do not elborte this point but refer to the stndrd literture on numericl qudrture. 4. Explicit Integrtion for the Cse c =0. For the cse c = 0, the integrl J(γ,c) in (3.4) cn be explicitly clculted s follows. We hve G p (r, 0) = 1 p rp nd consequently, see (3.3), = J(γ,0) = 1 b (θ) sin γ 2 (θ) R γ +1 γ +1 (θ) dθ = dγ +1 γ +1 b (θ) sin γ 2 (θ) cos γ +1 (θ β) dθ b β = dγ +1 (µ + β) sin γ 2 (µ + β) dµ. γ +1 β cos γ +1 (µ) For convenience, let us set A := sin(β) nd B := cos(β). Then (µ + β) sin γ 2 (µ + β) γ 1 p 1 =0 p 2 =0 cos γ +1 (µ) γ 2 ( γ1 ( 1) γ 2 p 2 p 1 )( γ2 p 2 ) sin p 1+p 2 (µ) cos γ p 1 p 2 (µ) A γ 2+p 1 p 2B γ 1 +p 2 p 1. cos γ +1 (µ)

7 From this we hve computtion of singulr integrls over tringles 7 (4.1) J(γ,0) = dγ +1 γ +1 γ 1 γ 2 p 1 =0 p 2 =0 where we denoted by F k the integrl F k := ( 1) γ 2 p 2 ( γ1 b β β Now integrtion by prts yields the recursion p 1 )( γ2 p 2 sin k (µ) cos k+1 (µ) dµ. ) A γ 2+p 1 p 2B γ 1 +p 2 p 1 Fp1 +p 2, ( ) 1 + tn(µ/2) µ=b β F 0 =ln, 1 tn(µ/2) µ= β µ=b β 1 F 1 = cos(µ), µ= β F k = 1 sin k 1 µ=b β (µ) k cos k k 1 (µ) k F k 2 for k =2, 3,..., which mkes (4.1) explicit. µ= β 5. Asymptotic Estimtes for Smll c. In colloction or Glerkin methods, the entries of the system mtrix i,j re typiclly pproximted only to such precision tht the order of the discretiztion error is not worsened. Hence, it is of interest to investigte the error which is introduced when replcing the integrl J(γ,c) byj(γ,0) for smll c. If this error is smll, then the (more efficient) explicit clcultions described in section 4 cn be used for pproximting J(γ,c). We differentite the recursion (3.6) twice with respect to c nd let c tend to zero. Thus the following limits re obtined: 2 G 1 (r, 0 ± 0) = 1, 2 G p (r, 0) = 0 for p =2, 3,..., 2 2 G p(r, 0) = rp 2 p 2 for p =3, 4,..., Here 2 indictes the prtil derivtive with respect to the c-vrible, nd 0 ± 0 indictes whether the limit c 0 is tken from the right or the left. As consequence, we obtin the following symptotic estimtes for the integrl in (3.4): O ( c ) for γ =0, J(γ,c) =J(γ,0) + o ( c ) for γ =1, O ( c 2) for γ =2, 3,...

8 8 e. l. llgower, k. georg nd k. klik Let us recll tht the prmeter c describes the distnce of the colloction point Q p from the plne generted by the tringle q. On the other hnd, the discretiztion error of the colloction or Glerkin method is usully known nd expressed s O(h τ ) where h indictes the mximl dimeter of the tringles q. Hence, if for exmple γ 2, then the pproximtion J(γ,c) J(γ,0) is permissible for colloction points Q p which re ner to q in the sense tht the distnce c does not exceed the order O ( h τ/2). REFERENCES [1] M. H. Alibdi, W. S. Hll nd T. T. Hibbs, Exct singulrity cncelling for the potentil kernel in the boundry element method, Communictions in Applied Numericl Methods, 3 (1987), [2] E.L. Allgower nd K. Georg, Numericl continution methods: An introduction. Springer Verlg, Berlin, Heidelberg, New York, 1990 [3] P. K. Bnerjee nd R. Butterfield, Boundry Element Methods in Engineering Science, [4] R. B Guenther nd J. Lee, Prtil differentil equtions of mthemticl physics nd integrl equtions, Prentice Hll, Englewood Cliffs, 1987 [5] W. Hckbusch, Integrlgleichungen. Theorie und Numerik, B. G. Teubner, Stuttgrt, 1989 [6] K. Hymi nd C. A. Brebbi, A new coordinte trnsformtion method for singulr nd nerly singulr integrls over generl curved boundry elements, Boundry Elements IX, vol 1, editors: C. A. Brebbi, W. L. Wendlnd, G. Kutiu, Springer Verlg, Berlin, Heidelberg, New York [7] M. A. Jswon nd G. T. Symm, Integrl Eqution Methods in Potentil Theory nd Electrosttics, Acdemic Press, New York, 1977 [8] E. Mrtensen, Potentiltheorie, B. G. Teubner, Stuttgrt, 1968 [9] Ch. Schwb nd W. L. Wendlnd, 3-D BEM nd numericl integrtion, in Proc. Seventh. Conf. on Boundry Element Methods in Engineering, C. Brebbi, ed., Springer, 1985 [10] W. L. Wendlnd, Asymptotic ccurcy nd convergence progress in boundry element methods, editor: C. A. Brebbi, Pentech-Press, London, Plymouth, (1981), [11] R. Widmnn, An efficient lgorithm for the tringultion of surfces in R 3, preprint, Colordo Stte University, October 1990.