Numerical Solution of Boundary Value Problems for the Laplacian in R 3 in the Case of Complex Boundary Surface

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1 Computational Applied Mathematics Jounal 5; (: 9-5 Published online Febuay 5 ( Numeical Solution of Bounday Value Poblems fo the Laplacian in R in the Case of Complex Bounday Suface Olex Polishchuk Depatment of Nonlinea Mathematical Analysis Pidstyhach Institute fo Applied Poblems of Mechanics Mathematics National Academy of Sciences of Ukaine Lviv Ukaine addess od_polishchuk@uk.net Keywods Bounday Integal Equations Pojection Methods Collocation Removal of Singulaity Received: Febuay 5 Revised: Febuay 5 Accepted: Febuay 5 Citation Olex Polishchuk. Numeical Solution of Bounday Value Poblems fo the Laplacian in R in the Case of Complex Bounday Suface. Computational Applied Mathematics Jounal. Vol. No. 5 pp Abstact Potential theoy is one of the ways to solve the bounday value poblems fo the Laplacian. Well-posed solvability of integal equations equivalent to bounday value poblems allow to use fo thei solution well known pojection methods. In many applied poblems the bounday sufaces have complex geomety contain the edges cone points. Togethe with the singulaity in the kenel this gives ise to a singulaity in the seached density of the potential. The methods ae poposed fo emoval of singulaities in the kenels unknown densities of potentials that significantly impove the accuacy of pojection methods as well as thei numeical solution.. Intoduction Duing modeling of many physical pocesses (diffusion heat flow electostatic field fictionless flow elastic motion of solids etc. the need fo solution of bounday value poblems fo the Laplacian in R aises []. In the case of complex geomety of the bounday suface fo the solution of such poblems should be used the potential theoy methods []. Depending on the popeties of envionment solution of bounday value poblem can be sought in the fom of simple laye potential [] double laye potential [4] o the sum of these potentials [5]. Systems of integal equations equivalent to the bounday value poblems may contain integal equations of the fist kind [-5]. Review of the conditions of well-posed solvability of these equations fo the simple double laye potentials closed bounday sufaces contained in [6] fo tied bounday sufaces in [7]. Review of the conditions of well-posed solvability of these equations fo the sum of potentials contained in [8]. These esults allow to use fo the the solution of systems of integal equations the well-known pojection methods [ 9-]. Implementation of these methods equies calculation of singulaities in kenel unknown density of potentials when appoaching the edge o the cone of the bounday suface which often occus in existing devices such as high voltage tansmission towes ada systems of vaious types etc. This poblem was studied in [4-9]. In the given aticles analytical methods fo calculation of singulaities fo paticula suface ae poposed. The use of finite element appoximation of unknown potential density povides additional oppotunities to calculate its singulaities build new algoithms fo calculation of matix coefficients of the system of linea algebaic equations discetized system integal equations significantly impoves the accuacy of thei numeical solution.

2 Computational Applied Mathematics Jounal 5; (: 9-5. Collocation Method Function of Singulaity of Potential Density Let N i be the bounday suface in R. Suppose that on each i suface i is given bounday condition γ j iu( x fj i( x x i i N j M which must be satisfied the seached hamonic function u (x x R. Tace opeato γ j i of function u (x on suface i detemines [] the type of the bounday condition (Diichlet Neumann Poincae jump unknown function o ( its nomal deivative etc. f j i sets the value of this condition. The bounday conditions can be as one-sided double-sided [6-8]. They may also be mixed when on diffeent sides of the bounday suface set the bounday conditions of diffeent types [5]. Denote whee u( x W( σ q( x ( U V( σ q ( x G( x x R x y σ ( ( U ( x σ( G( x dy 4 σ is a simple laye potential y is a density of simple laye potential Q( x n ( Vq ( x q( Q( x dy 4 x y x R is a double laye potential y q y is a density of double laye potential. To ( detemine the unknown densities of the potentials need to solve the system of integal equations N k γ j kw q x (σ ( f j i( x x i ( i N j M. Well-known pojection methods (collocation Galekin least squaes etc. can be used to solve the system (. These methods ae well investigated fo stability convegence commonly used to solve many poblems of mathematical physics [-4 4]. Let Si [ ai] [ bi] R be the ectangula domain of paametes of suface i in which a ectangula gid S h i with steps τ h ν i bi / Nν i i N is h i ai/ N i N geneated. Let Φ ( i Nν i N τ i N { ϕ ν i lm( } l m τ [ ai] ν [ bi] be the system of piecewise-linea finite elements defined on S h i. Conside collocation method T fo solution of the system ( when the set of points of obsevation coincides with the nodes of the gid S h i. System of collocation equations in this case has the fom N k Nν k N k Nν k N γ j kw φlm φɶ ɶ x k lm pi ti l ɶ k mk lk mɶ k f ( x x whee a lm j i pi ti pi ti i ( ( ( ( p i ( N i t i ( Nν i i N j M. ( b lm ae unknown paametes xp i t i i i N is the set of points of obsevation. Conditions of stability convegence of the method ( defined in []. The main facto affecting the accuacy of appoximate solution of the system of collocation equations is the calculation of integals with singulaities in thei matices coefficients. Let P τ ] ν ] be the cetainelement [ τ [ ν of the gid S h i τ τ h τ i ν ν h ν i i N. Elimination of singulaity of the density is only suitable fo gid elements that ae tangent to the edges o cones of the bounday suface. Moeove constuction of the function of singulaity fo domain of paametes S i is often a difficult poblem e.g. fo sufaces with cuved cuts. Conside the elements of the matix of the system of collocation equations that ae fomed fom the integals depending on the given bounday conditions: ρ( J( I dνdτ x( ( y I ρ( J( dνdτ (4 n x( yɶ I ρ( J( dνdτ n x( yɶ I4 ρ( J( dνdτ (6 n n x( yɶ whee ρ ( is the function of singulaity of potential density L ( is a basic function defined on P J( is the Jacobian of coodinate tansfomation. Constuct the function of singulaity of potential density ρ ( τ τ ν ν τ τ τ τ ν ν τ τ yɶ yɶ x x 4 ν ν ν ν Usually mes P<< mes Si i.e. no moe than two indexes j j 4 may be diffeent fom zeo. If P is intenal element of the gid S h i (5 i N point of obsevation.

3 Olex Polishchuk: Numeical Solution of Bounday Value Poblems fo the Laplacian in R in the Case of Complex Bounday Suface does not belong to P then function unde the integal in (- (6 is smooth fo numeical integation the Gauss quadatue fomula with constant weight vaious degees of accuacy [] is used.. Removal of the Singulaity in the Density Kenel of Simple Laye Potential In ode to simplify the pesentation we assume that [ ] [] is the unit squae on the plane x y. Paametic epesentation of such suface is given by x x τ ν ν x Jacobian of coodinate tansfomation J (. If P is the bode element of the gid e.g. τ ν point of obsevation does not belong to P then integal I has the fom I ρ( J( d ν d τ x( yɶ h h dν dτ. x( yɶ In ode to calculate I sequentially the quadatue fomula poposed in [] is used: ϕ( τ τ τ γ d ( γ ( + N N ( tk + RN k φ (7 (γ k whee t k R N k N. N + (N + ( γ Choice of the fomula (7 is explained by the simplicity of dependence on the singulaity ode γ. Let P be the intenal element of the gid ɶy ( y y y ( τ ν. In this case τν I ( τ τ + ( ν ν dνdτ. (8 Implement the change of vaiables in (8 by tansition to pola coodinates system with cente in the point ( τ ν Then integal τ τ + cosθ ν ν + sinθ. (9 ( I τ( d whee ( is function which detemines the distance fom the obsevation point to the opposite sides of the element has no singulaity. Fo its calculation the Gauss quadatue fomula with constant weight is used. If P is the bode element of instead of (8 we have τ ν I in the case /[(( τ τ + ( ν ν ( τ τ + ( ν ν ]dνdτ ( τ τ( ν L ( ( ( τ τ( ν we obtain τν τ τ ν ν I h τ h ( ( ν τν ( τ τ + ( ν Using the change of vaiables (9 we obtain h h dνdτ. I cos θsin θ ( ( Function unde the integal in ( has no singulaity. To calculate it Gauss quadatue fomula with constant weight is used. If we obtain ( τ τ ( ( ν ν L τ ι ( ( τ τ( ν cos θ sin θ I ( ( cos sin ( θ θ θ The fist integal in ( is calculated by dividing the inteval of integation into two intevals [ ε ] [ ε ] ε 8 eplacing sin θ θ on [ ε ]. Appoximate value of the integal with the singulaity is calculated using the quadatue fomula (7 othe integals by means of Gauss quadatue fomula with constant weight. Do the same fo ( τ τ( ν L (. (4 ( τ τ ( ν ν In this case

4 Computational Applied Mathematics Jounal 5; (: 9-5 sin θ cos θ I ( (5 cos sin ( θ θ θ The fist integal in (5 is calculated by dividing the inteval of integation into two intevals [ ε] [ ε ] ε 8 eplacing cos θ on [ ε ] (. Appoximate value of the integal with the singulaity is calculated using the quadatue fomula (7 othe integals by means of Gauss quadatue fomula with constant weight. In case we obtain ( τ τ ( ( ν ν L τ ι (6 ( τ τ( ν ( [ θ cos θsin I h h ( h cos θ h sin θ ( + sin θ cos θ ν τ h h ( + ] ( cos θsin θ θ (7 Fo the calculation of integals with singulaities in (7 the same techniques as fo ( (5 ae used. Similaly the emoval of the singulaity in the density of potential if the element of the gid is tangent to the edge of is pefomed. 4. Removal of the Singulaity in the Density Nomal Deivative of the Kenel of Simple Laye Potential Conside integal (4 if Pis the intenal element of the gid S h ( τ is the obsevation point. In this case I [( τ τcos ɶ + ( ν cos ɶ β] (( τ τ + ( d d ν ν ν τ (8 whee is the angle between the vectos is the angle between the vectos between the vectos e y e y e y n y β n y the angle n y fo the suface is equal to zeo. Having changed vaiables (9 in integal (8 we obtain ( I τ( ψ( θ β d whee ψ( θ β cos cosθ + cos β sinθ. Then in case ( we have I θ θ ψ θ β h cos sin ( ν In case ( we obtain ɶ ɶ ν ( θ. (9 I cos θ ψ( θ β( h ( ( sin ( In case (4 we obtain ɶ ɶ τ I sin θ ψ( θ β( h ( ( cos ( Functions unde the integals in (9 ( ae smooth fo calculation of integals the Gauss quadatue fomula with constant weight is used. Finally in the case (6 we obtain h τ I ψ( θ β( ( cosθsin θ ( ( ( cosθ + sin ψ( θ β d ( Take the function q ( θ δ ln(( ( δ / δ whee paamete δ << ( θ as an appoximate value of the ( integal d. If P is the bounday element of the gid S h e.g. τ ν then instead of (9 ( we have integals h h I ( cos θsin θ ψ( θ ɶ ɶ β (

5 Olex Polishchuk: Numeical Solution of Bounday Value Poblems fo the Laplacian in R in the Case of Complex Bounday Suface cos θ sin θ I ( ψ( θ ɶ ɶ β ( cos θsin θ ψ( θ ɶ ɶ β (4 sin θ cos θ I ( ψ( θ ɶ ɶ β ( cos θsin θ ψ( θ ɶ ɶ β (5 Integals with singulaities in (4 (5 ae calculated similaly to ( (5. Instead of ( we have I ( cos θ sin θ ψ( θ ɶ ɶ β sin cos τ θ ν h h h h θ ( ( + ψ( θ ɶ ɶ β cos θ sin θ ( d cos sin h h θ θ (6 Take the function ɶq ( θ ɶ δ ( ( ɶ δ /( + ɶ δ ( θ whee paamete δ << ( θ as an appoximate value of the ( d integal. + + The emoval of the singulaity in the density of potential if the element of the gid is tangent to the edge of is pefomed similaly. 5. Removal of the Singulaity in the Density Kenel of Double Laye Potential Conside the integal (5. In this case I [( τ τcos + ( ν cos β] (( τ τ + ( d d ν ν ν τ whee is the angle between the vectos is the angle between the vectos ex (7 e x n x the angle n x β between the vectos e x nx fo the suface is equal zeo. Having changed vaiables (9 in integal (7 we obtain ( I τ( φ( θ β d (8 whee φ( θ β coscosθ + cos βsinθ. In ode to calculate the integal (8 the same methods as in pevious subsection ae used. Similaly the integal I 4 educes to the fom whee ( I4 ( ( ( ɶ ɶ/ dd τ θ ν θ χ θ β β θ(9 χ( θ β ɶ ɶ β (cos θ sin coscosɶ + cosθsinθcoscosɶ β + cosθsinθcos βcosɶ + (sin θ cos cos βcosɶ β In ode to calculate the integal (9 the quadatue fomulas fo singula integals poposed in [] is used. We can also use the Galekin method fo numeical solution of coesponding integal equations although it involves significant incease in the computational expenditues. Let be the pat of the bounday suface with paametic epesentation x x( x x( x x( τ a ν b ectangula gid S h be geneated in domain of paametes S [ a] [ b] R. Denote R ( x / ( y ν P y is a fixed point of element of the gid P. In case y x( ν the integal ( has singulaity. To emove the latte make the change of

6 Computational Applied Mathematics Jounal 5; (: vaiables (9 whee τ ν ae paametic coodinates of the point y. Then R ( dτdν R ( τ( d K ( d when the integal ( can be singula only due to potential density. The latte is easy emoved when the density is appoximated by the piecewise-linea elements the quadatue fomula (7 is used to calculate (. In geneal the eduction of integal ( to a single integal with a eplacement (9 applied fo the flat pat of the bounday suface is impossible. Theefoe to calculate the integal ( I f ( dd θ consistently the Gauss quadatue fomula one of the peviously consideed methods fo the calculation of singula integals which depends on the value of ae used. If step of the gid on the suface is small enough then element P S h can be eplaced by a flat. In this case we can use algoithms consideed above fo emoval of singulaities fo the flat suface. Poposed appoach can be easily extended to the case of nonectangula gids highe-ode finite element appoximations othe pojection methods fo numeical solution of the system of integal equations (. 6. Results Discussion Conside seveal examples with known analytical solution that ae used to study numeical solution distotions that occu close to edges cone points of bounday suface. They allow to examine the effectiveness of application of methods poposed fo singulaities emoval in kenels densities of potentials. Let G [ ] [ ] [ ] be the unit cube G whee is a flat tied suface [.5.75] [.5.75] which lies in the plane y. 5. Assume that the unknown function is continuous when cossing G. In ode to illustate the effectiveness of the poposed algoithms the following bounday value poblems wee solved. Poblem. Find function u G u in G which satisfied bounday condition u. Solution of this poblem is known namely u in G. In all pats of (the sides of the cube the suface the ectangula gid with h τ.5 was geneated. The solution was poposed in the fom of simple laye potential with a piecewise linea appoximation of the smooth pat of the unknown potential density. The values i i 4 ae equal to.5. Fo the solution of integal equation the collocation method is used with the obsevation points coinciding with the gid nodes. The dimension of the system of collocation equations N 458. To calculate integals ( fist only Gauss quadatue fomula of the fouth ode with constant weight without emoval of singulaities was used. The accuacy of the appoximate solution in G when appoaching the middle of the edge of the cube was equal.% when appoaching the middle of the edge.5% when appoaching the cone points of G 5.8%. When Gauss quadatue fomula of the fouth ode with constant coefficients as well as poposed algoithms fo emoving singulaities wee used the accuacy of appoximate solution in G while appoaching the middle of the cube edge was equal to. % when appoaching the middle of the edge.5 % when appoaching the cone points of bounday suface.9 %. Result of simila accuacy may be achieved without usage of poposed algoithms fo emoval singulaities. Fo this the gid with step h τ.65 has to geneated on the bounday suface. The dimension of the system of collocation equations in this case N 74. The numbe of opeations equied fo the fomation of the system of collocation equations in second case is ove 4 times highe than numbe of opeations equied to geneate appopiate system in fist case. The numbe of opeations equied fo solution of the system of collocation equations with the use of Gauss method O ( N is ove 5 times highe than numbe of opeations O ( N equied fo solution of the system of linea algebaic equations in case of singulaities emoval. Poblem diffes fom poblem only in bounday value condition on the suface which is assumed to be u n. To calculate integals ( (4 fist only Gauss quadatue fomula of the fouth-ode with constant weight without singulaities emoval was used. The accuacy of the appoximate solution in G when appoaching the middle of the edge of the cube was equal.% when appoaching the middle of the edge 5.7% when appoaching the cone points of G 8.%. When Gauss quadatue fomula of the fouth ode with constant coefficients as well as poposed algoithms fo emoving singulaities wee used fo integals ( (4 the accuacy of appoximate solution in G while appoaching the middle of the cube edge was equal.6 % when appoaching the middle of the edge.9% when appoaching the cone points of G.%. 7. Conclusions Bounday integal equation method is a poweful tool fo solving many poblems of mathematical physics. The accuacy of the numeical solution of these equations depends essentially on the pesence of singulaities in the kenel density of potential. The algoithms ae developed fo emoval singulaities in densities kenels of simple double laye potentials its nomal deivatives nea the edges cone points of the bounday suface using the popeties of finite element appoximation. Numeical

7 5 Olex Polishchuk: Numeical Solution of Bounday Value Poblems fo the Laplacian in R in the Case of Complex Bounday Suface expeiments demonstate the high efficiency of appoach poposed. These methods can be used to impove the accuacy of the numeical solution of integal integaldiffeential equations with singulaities equivalent to many othe poblems of mathematical physics. Refeences [] T. V. Homadka II C. Lay The complex vaiable bounday elements method in engineeing analysis Spinge-Velag New Yok USA 987. [] G. C. Hsiao W. L. Wendl Bounday integal equations Spinge Belin Gemany 8. [] J. C. Nedelec J. Planchad Une methode vaiationnelle d elements finis pou la esolution numeique d un pobleme exteieu dans R R.A.I.R.O. R no.7 pp [4] J. Gioue Fomulation vaiationnelle pa equations integales de poblemes aux limites exteieus Rappot Intene du Cente de mathematiques Appliquees de l Ecole Politechnique no [5] A.D. Polishchuk Solution of the double-sided Diichlet- Neumann poblem fo the Laplacian in R by means of the potential theoy methods Mathematical Methods Physicomechanical Fields vol. 48 no. pp [6] A. D. Polishchuk Constuction of bounday opeatos fo the Laplacian Poceedings of the 7th Intenational Semina Diect Invese Poblems of Electomagnetic Acoustic Wave Theoy Lviv Ukaine Septembe -5 5 pp [7] A. D. Polishchuk Constuction of bounday opeatos fo the Laplacian in the case of tied bounday suface Poceedings of the th Intenational Semina Diect Invese Poblems of Electomagnetic Acoustic Wave Theoy Tbilisi Geogia - Octobe 6 pp.5-6. [8] A. D. Polishchuk Solution of double-sided bounday value poblems fo the Laplacian in R by means of potential theoy methods Poceedings of the 7th Intenational Semina Diect Invese Poblems of Electomagnetic Acoustic Wave Theoy Tbilisi Geogia Septembe -5 4 pp [9] P. Kythe P. Pui Computational Methods fo Linea Integal Equations Spinge Science & Business Media. [] A. G. Ramm A collocation method fo solving integal equations Intenational Jounal of Computing Science Mathematics vol. no. pp [] J. Rashidinia M. Zaebnia Convegence of appoximate solution of system of Fedholm integal equations Jounal of Mathematical Analysis Applications vol. no. pp [] K. Maleknejad N. Aghazadeh R. Mollapouasl Numeical solution of Fedholm integal equation of the fist kind with collocation method estimation of eo bound Applied Mathematics Computation vol. 79 no. pp [] A. D. Polishchuk About numeical solution of some integal equations of the fist kind Poceedings of the 7th Intenational Semina Diect Invese Poblems of Electomagnetic Acoustic Wave Theoy Tbilisi Geogia 4-7 Septembe pp [4] A. I. Gebennikov Methods of spline-collocation double spline-appoximation fo solution of opeato equations its application to solution of integal equations with singulaities In: Methods algoithms in numeical analysis Nauka Moscow USSR pp [5] A. G. Kyukchan N. I. Sminova Solution of wave diffaction poblems by the method of continued bounday conditions Acoustic Jounal vol. 5 no. 4 pp [6] G. Fiсhea Asymptotic behavio of the electic field density of the electic chage in the neighbohood of singula points of a conducting suface Russian Mathematical Suveys vol. no. pp [7] A. V. Amenitskiy Bounday integal equations fo nanlizing dynamic poblems of -D poouseelasticity Poblems of Stenght Plasticity vol. pp [8] A. V. Ivanov M. A. Tiunov Use of bounday integal equations method duing modeling of fomation dynamic of sheaf in installation of the electonic cooling Bulletin of Novosibisk State Univesity Physics vol. no. pp [9] M. Y. Medvedyk Collocation method fo solution of the poblem of diffaction of electomagnetic waves by a dielectic body located on the esonato Bulletin of High School Volga Region Physics Mathematics vol. pp.8-4. [] H. Tiebel Theoy of function spaces Academische Velagsgesellschaft Postig K.-G. Leibzig Gemany 98. [] V. I. Kylov Appoximate calculation of integals The Macmillan Co New Yok USA 96. [] I. V. Boikov Asymptotically optimal algoithms fo calculation of singula integals In: Use of numeical methods in scientific investigations Penza state Univesity Pess Penza USSR pp [] H. V. Stoll P. Stauss On the numeical integation of cetain singula integals Computing vol. 48 pp