https://oldena.lpnu.ua/handle/ntb/44891| Title: | Finite element approximations in projection methods for solution of some Fredholm integral equation of the first kind |
| Other Titles: | Скінченно-елементні апроксимації у проекційних методах розв’язання деяких інтегральних рівнянь Фредгольма першого роду |
| Authors: | Поліщук, О. Polishchuk, O. |
| Affiliation: | Інститут прикладних проблем механіки і математики ім. Я. С. Підстригача Pidstryhach Institute for Applied Problems of Mechanics and Mathematics |
| Bibliographic description (Ukraine): | Polishchuk O. Finite element approximations in projection methods for solution of some Fredholm integral equation of the first kind / O. Polishchuk // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2018. — Vol 5. — No 1. — P. 74–87. |
| Bibliographic description (International): | Polishchuk O. Finite element approximations in projection methods for solution of some Fredholm integral equation of the first kind / O. Polishchuk // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2018. — Vol 5. — No 1. — P. 74–87. |
| Is part of: | Mathematical Modeling and Computing, 1 (5), 2018 |
| Journal/Collection: | Mathematical Modeling and Computing |
| Issue: | 1 |
| Volume: | 5 |
| Issue Date: | 15-Jan-2018 |
| Publisher: | Lviv Politechnic Publishing House |
| Place of the edition/event: | Lviv |
| UDC: | 519.6 |
| Keywords: | потенцiал iнтегральне рiвняння коректна розв’язнiсть лагран- жева апроксимацiя В-сплайн метод Гальоркiна метод колокацiї збiжнiсть potential integral equation well-posed solvability B-spline Lagrange interpolation Galerkin method collocation method convergence |
| Number of pages: | 14 |
| Page range: | 74-87 |
| Start page: | 74 |
| End page: | 87 |
| Abstract: | Дослiджено апроксимацiйнi властивостi В-сплайнiв та лагранжевих кiнцевих елемен-
тiв у гiльбертових просторах функцiй, визначених на поверхнях у тривимiрному про-
сторi. Встановлено умови збiжностi методiв Гальоркiна та колокацiї розв’язання iн-
тегрального рiвняння Фредгольма першого роду для потенцiалу простого шару, еквi-
валентного задачi Дiрiхле для рiвняння Лапласа в R3. Визначено оцiнку похибки
наближеного розв’язку цiєї задачi, отриманого за допомогою методiв теорiї потенцiа-
лу. Approximation properties of B-splines and Lagrangian finite elements in Hilbert spaces of functions defined on surfaces in three-dimensional space are investigated. The conditions for the convergence of Galerkin and collocation methods for solution of the Fredholm integral equation of the first kind for the simple layer potential that is equivalent to the Dirichlet problem for Laplace equation in R3 are established. The estimation of the error of approximate solution of this problem, obtained by means of the potential theory methods, is determined. |
| URI: | https://ena.lpnu.ua/handle/ntb/44891 |
| Copyright owner: | © 2018 Lviv Polytechnic National University CMM IAPMM NASU © 2018 Lviv Polytechnic National University CMM IAPMM NASU |
| References (Ukraine): | [1] HromadkallT.V., LaiC. The complex variable boundary elements method in engineering analysis. Springer-Verlag (1987). [2] HsiaoG.C., WendlandW. L. Boundary integral equations. Springer (2008). [3] HackbuschW. (Ed.) Integral equations: theory and numerical treatment (Vol. 120). Birkh¨auser Basel (2012). [4] KytheP.K., PuriP. Computational methods for linear integral equations. Springer Science & Business Media (2011). [5] McLeanW.C.H. Strongly elliptic systems and boundary integral equations. Cambridge University Press (2000). [6] GolbergM.A. (Ed.) Numerical solution of integral equations (Vol. 42). Springer Science & Business Media (2013). [7] Il’inV.P., PolishchukA.D. About numerical solution of dimensional problems of the potential theory. Variation-difference methods in the problems of numerical analysis. 3, 28–44 (1987), (in Russian). [8] RjasanowS., SteinbachO. The fast solution of boundary integral equations. Springer Science & Business Media (2007). [9] Nedelec J.-C., Planchard J. Une m´ethode variationnelle d’´el´ements finis pour la r´esolution num´erique d’un probl`eme ext´erieur dans R3. Revue fran¸caise d’automatique, informatique, recherche op´erationnelle. Math´ematique. 7 (R-3), 105–129 (1973). [10] PolishchukA.D. Simple and double layer potentials in the Hilbert spaces. Proceedings of 8th International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and AcousticWave Theory, DIPED-2003. 94–97 (2003). [11] Giroure J. Formulation variationnelle par equations integrales de problemes aux limites ext´erieurs. Rapport Interne du Centre de Math´ematiques Appliqu´ees de l’´ Ecole Polytechnique. 6, 97 (1976). [12] PolishchukO.D. Solution of Neumann problem for the Laplacian in R3 for tired surface by means of double layer potential. Volyn’ Mathematical Bulletin. 2 (11), 61–64 (2004), (in Ukrainian). [13] PolishchukO.D. Solution of bilateral Dirichlet and Neumann problems for the Laplacian in R3 for tired surface by means of the potential theory methods. Applied Problems of Mechanics and Mathematics. 2, 80–87 (2004). [14] PolishchukO.D. Solution of bilateral Dirichlet–Neumann problems for the Laplacian in R3 by potential theory methods. Mathematical Methods and Physicomechanical Fields. 48 (1), 59–64 (2005), (in Ukrainian). [15] Baldino P.R. An integral equation solution of the mixed problem for the Laplacian in R3. Rapport Interne du Centre de Math´ematiques Appliqu´ees de l’´ Ecole Polytechnique. 48 (1979). [16] Li Z.C., Huang J., HuangH.-T. Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace’s equation. Computing. 88 (1–2), 1–29 (2010). [17] LiangX.-Z., LiuM.-C., Che X.-J. Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets. Journal of Computational and Applied Mathematics. 136 (1), 149–161 (2001). [18] PolishchukA.D. Construction of boundary operators for the Laplacian. 1. Using of simple layer potential. Proceedings of Xth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED-2005. 137–142 (2005). [19] PolishchukA.D. Construction of boundary operators for the Laplacian in the case of tired boundary surface. - I. Using the Simple Layer Potential. Proceedings of XIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED-2006. 153–156 (2006). PolishchukA.D. Construction of boundary operators for the Laplacian in the case of tired boundary surface. - II. Using the Double Layer Potential Proceedings of XIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED-2006. 157–160 (2006). [20] PolishchukA.D. Solution of double-sided boundary value problems for the Laplacian in R3 by means of potential theory methods. Proceedings of International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED-2014. 140–142 (2014). [21] Chen Z., XuY., YangH. Fast collocation methods for solving ill-posed integral equations of the first kind. Inverse Problems. 24 (6), 065007 (2008). [22] NairM.T., Pereverzev S.V. Regularized collocation method for Fredholm integral equations of the first kind. Journal of Complexity. 23 (4), 454–467 (2007). [23] Chen Z., Cheng S., NelakantiG., YangH. A fast multiscale Galerkin method for the first kind ill-posed integral equations via Tikhonov regularization. International Journal of Computer Mathematics. 87 (3),565–582 (2010). [24] PolishchukA.D. About numerical solution of potential theory integral equations. Preprint, Computer centre of Siberian Division of AS of the USSR, 743 (1987). [25] PolishchukA.D. About convergence the methods of projections for solution potential theory integral equation. Preprint, Computer centre of Siberian Division of AS of the USSR, 776 (1988). [26] ReinhardtH. J. Analysis of approximation methods for differential and integral equations (Vol. 57). Springer Science & Business Media (2012). [27] MaleknejadK., DeriliH. Numerical solution of integral equations by using combination of Splinecollocation method and Lagrange interpolation. Applied Mathematics and Computation. 175 (2), 1235–1244 (2006). [28] PolishchukO. Numerical solution of boundary value problems for the Laplacian in R3 in case of complex boundary surface. Computational and Applied Mathematics Journal. 1 (2), 29–35 (2015). [29] Aubin J.-P. Approximation of elliptic boundary value problems. Wiley-Interscience (1972). [30] Lions J. L., Magenes E. Probl`emes aux limites non homog`enes et applications. Dunod (1968). [31] PetryshynW.V. Constructional proof of Lax-Milgram lemma and its application to non-K-P.D. Abstract and differential operator equations. SIAM Journal Numerical Analysis. 2 (3), 404–420 (1965). [32] AleksidzeM.A. Solution of boundary value problems by means of decomposition on orthogonal functions. Nauka, Moscow (1978), (in Russian). |
| References (International): | [1] HromadkallT.V., LaiC. The complex variable boundary elements method in engineering analysis. Springer-Verlag (1987). [2] HsiaoG.C., WendlandW. L. Boundary integral equations. Springer (2008). [3] HackbuschW. (Ed.) Integral equations: theory and numerical treatment (Vol. 120). Birkh¨auser Basel (2012). [4] KytheP.K., PuriP. Computational methods for linear integral equations. Springer Science & Business Media (2011). [5] McLeanW.C.H. Strongly elliptic systems and boundary integral equations. Cambridge University Press (2000). [6] GolbergM.A. (Ed.) Numerical solution of integral equations (Vol. 42). Springer Science & Business Media (2013). [7] Il’inV.P., PolishchukA.D. About numerical solution of dimensional problems of the potential theory. Variation-difference methods in the problems of numerical analysis. 3, 28–44 (1987), (in Russian). [8] RjasanowS., SteinbachO. The fast solution of boundary integral equations. Springer Science & Business Media (2007). [9] Nedelec J.-C., Planchard J. Une m´ethode variationnelle d’´el´ements finis pour la r´esolution num´erique d’un probl`eme ext´erieur dans R3. Revue fran¸caise d’automatique, informatique, recherche op´erationnelle. Math´ematique. 7 (R-3), 105–129 (1973). [10] PolishchukA.D. Simple and double layer potentials in the Hilbert spaces. Proceedings of 8th International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and AcousticWave Theory, DIPED-2003. 94–97 (2003). [11] Giroure J. Formulation variationnelle par equations integrales de problemes aux limites ext´erieurs. Rapport Interne du Centre de Math´ematiques Appliqu´ees de l’´ Ecole Polytechnique. 6, 97 (1976). [12] PolishchukO.D. Solution of Neumann problem for the Laplacian in R3 for tired surface by means of double layer potential. Volyn’ Mathematical Bulletin. 2 (11), 61–64 (2004), (in Ukrainian). [13] PolishchukO.D. Solution of bilateral Dirichlet and Neumann problems for the Laplacian in R3 for tired surface by means of the potential theory methods. Applied Problems of Mechanics and Mathematics. 2, 80–87 (2004). [14] PolishchukO.D. Solution of bilateral Dirichlet–Neumann problems for the Laplacian in R3 by potential theory methods. Mathematical Methods and Physicomechanical Fields. 48 (1), 59–64 (2005), (in Ukrainian). [15] Baldino P.R. An integral equation solution of the mixed problem for the Laplacian in R3. Rapport Interne du Centre de Math´ematiques Appliqu´ees de l’´ Ecole Polytechnique. 48 (1979). [16] Li Z.C., Huang J., HuangH.-T. Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace’s equation. Computing. 88 (1–2), 1–29 (2010). [17] LiangX.-Z., LiuM.-C., Che X.-J. Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets. Journal of Computational and Applied Mathematics. 136 (1), 149–161 (2001). [18] PolishchukA.D. Construction of boundary operators for the Laplacian. 1. Using of simple layer potential. Proceedings of Xth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED-2005. 137–142 (2005). [19] PolishchukA.D. Construction of boundary operators for the Laplacian in the case of tired boundary surface, I. Using the Simple Layer Potential. Proceedings of XIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED-2006. 153–156 (2006). PolishchukA.D. Construction of boundary operators for the Laplacian in the case of tired boundary surface, II. Using the Double Layer Potential Proceedings of XIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED-2006. 157–160 (2006). [20] PolishchukA.D. Solution of double-sided boundary value problems for the Laplacian in R3 by means of potential theory methods. Proceedings of International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED-2014. 140–142 (2014). [21] Chen Z., XuY., YangH. Fast collocation methods for solving ill-posed integral equations of the first kind. Inverse Problems. 24 (6), 065007 (2008). [22] NairM.T., Pereverzev S.V. Regularized collocation method for Fredholm integral equations of the first kind. Journal of Complexity. 23 (4), 454–467 (2007). [23] Chen Z., Cheng S., NelakantiG., YangH. A fast multiscale Galerkin method for the first kind ill-posed integral equations via Tikhonov regularization. International Journal of Computer Mathematics. 87 (3),565–582 (2010). [24] PolishchukA.D. About numerical solution of potential theory integral equations. Preprint, Computer centre of Siberian Division of AS of the USSR, 743 (1987). [25] PolishchukA.D. About convergence the methods of projections for solution potential theory integral equation. Preprint, Computer centre of Siberian Division of AS of the USSR, 776 (1988). [26] ReinhardtH. J. Analysis of approximation methods for differential and integral equations (Vol. 57). Springer Science & Business Media (2012). [27] MaleknejadK., DeriliH. Numerical solution of integral equations by using combination of Splinecollocation method and Lagrange interpolation. Applied Mathematics and Computation. 175 (2), 1235–1244 (2006). [28] PolishchukO. Numerical solution of boundary value problems for the Laplacian in R3 in case of complex boundary surface. Computational and Applied Mathematics Journal. 1 (2), 29–35 (2015). [29] Aubin J.-P. Approximation of elliptic boundary value problems. Wiley-Interscience (1972). [30] Lions J. L., Magenes E. Probl`emes aux limites non homog`enes et applications. Dunod (1968). [31] PetryshynW.V. Constructional proof of Lax-Milgram lemma and its application to non-K-P.D. Abstract and differential operator equations. SIAM Journal Numerical Analysis. 2 (3), 404–420 (1965). [32] AleksidzeM.A. Solution of boundary value problems by means of decomposition on orthogonal functions. Nauka, Moscow (1978), (in Russian). |
| Content type: | Article |
| Appears in Collections: | Mathematical Modeling And Computing. – 2018. – Vol. 5, No. 1 |
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| 2018v5n1_Polishchuk_O-Finite_element_approximations_74-87.pdf | 1.49 MB | Adobe PDF | View/Open | |
| 2018v5n1_Polishchuk_O-Finite_element_approximations_74-87__COVER.png | 448.41 kB | image/png | View/Open |
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