Splitting extrapolation algorithms for solving the boundary integral equations of anisotropic Darcy’s equation on polygons by mechanical quadrature methods
In this paper we study the stability and convergence of the solution for the first kind integral equations of the anisotropic Darcy’s equations by the mechanical quadrature methods on closed polygonal boundaries in ℝ 2 . Using the collectively compact theory, we construct numerical solutions which c...
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| Abstract | In this paper we study the stability and convergence of the solution for the first kind integral equations of the anisotropic Darcy’s equations by the mechanical quadrature methods on closed polygonal boundaries in ℝ
2
. Using the collectively compact theory, we construct numerical solutions which converge with the order
, where
is the mesh size. In addition, An a posteriori asymptotic error representation is derived by splitting extrapolation methods in order to construct self-adaptive algorithms, and the convergence rate
can be achieved after using the splitting extrapolation methods once. Finally, the numerical examples show the efficiency of our methods. |
|---|---|
| AbstractList | In this paper we study the stability and convergence of the solution for the first kind integral equations of the anisotropic Darcy’s equations by the mechanical quadrature methods on closed polygonal boundaries in ℝ
2
. Using the collectively compact theory, we construct numerical solutions which converge with the order
, where
is the mesh size. In addition, An a posteriori asymptotic error representation is derived by splitting extrapolation methods in order to construct self-adaptive algorithms, and the convergence rate
can be achieved after using the splitting extrapolation methods once. Finally, the numerical examples show the efficiency of our methods. In this paper we study the stability and convergence of the solution for the first kind integral equations of the anisotropic Darcy’s equations by the mechanical quadrature methods on closed polygonal boundaries in ℝ2. Using the collectively compact theory, we construct numerical solutions which converge with the order , where is the mesh size. In addition, An a posteriori asymptotic error representation is derived by splitting extrapolation methods in order to construct self-adaptive algorithms, and the convergence rate can be achieved after using the splitting extrapolation methods once. Finally, the numerical examples show the efficiency of our methods. |
| Author | Wang, Chuan-Long Huang, Jin Luo, Xin |
| Author_xml | – sequence: 1 givenname: Xin surname: Luo fullname: Luo, Xin email: luoxin919@163.com organization: School of Mathematical Sciences, University of Electronic Science and Technology of China – sequence: 2 givenname: Jin surname: Huang fullname: Huang, Jin organization: School of Mathematical Sciences, University of Electronic Science and Technology of China – sequence: 3 givenname: Chuan-Long surname: Wang fullname: Wang, Chuan-Long organization: Department of Mathematics, Taiyuan Normal University |
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| CitedBy_id | crossref_primary_10_1016_j_cam_2020_113339 crossref_primary_10_1016_j_cam_2018_05_065 crossref_primary_10_1155_2020_6484890 crossref_primary_10_1080_00207160_2015_1095888 |
| Cites_doi | 10.1023/B:BITN.0000014568.20710.77 10.1080/00036818408839520 10.1090/S0025-5718-1991-1052084-0 10.56021/9780801858567 10.1137/080740763 10.1023/A:1006529122626 10.1016/j.enganabound.2010.07.004 10.1080/00207160701458252 10.1216/jiea/1181075650 10.1216/JIE-1988-1-4-517 10.1016/j.jcp.2009.08.014 10.1017/CBO9780511626340 10.1007/BF01061258 10.1142/2708 10.1093/imanum/8.1.105 10.1016/j.jcp.2008.09.011 10.1090/S0025-5718-1990-0995213-6 10.1016/j.jcp.2008.09.003 10.1016/j.apnum.2009.06.006 10.1016/j.enganabound.2006.01.005 10.1016/j.crma.2009.06.016 10.1023/A:1018989523557 10.1017/S033427000000429X 10.1016/0168-874X(94)90017-5 10.1114/1.1540103 10.1093/imanum/12.2.167 10.1016/j.cam.2011.01.019 10.1016/S0955-7997(00)00050-3 10.1007/s002110050107 10.1016/S0955-7997(97)00103-3 |
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| Keywords | Anisotropic Mechanical quadrature methods Darcy’s equation A posteriori estimate Splitting extrapolation algorithm |
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Anal.19841818321976750010.1080/00036818408839520 CostabelMErvinVJStephanEPOn the convergence of collocation methods for Symm’s integral equation on open curvesMath. Comput.1988511671799421480655.65144 ChandlerGAGalerkin’s method for boundary integral equations on polygonal domainsJ. Austral. Math Soc. Ser. B.1984261137505510559.6508310.1017/S033427000000429X YanYSloanIOn integral equations of the first kind with logarithmic kernelsJ. Integral Equations Appl.19881517548100840610.1216/JIE-1988-1-4-549 YanYThe collocation method for first-kind boundary integral equations on polygonal regionsMath. Comput.1990541391540685.6512110.1090/S0025-5718-1990-0995213-6 HuangJLüTLiZCThe mechanical quadrature methods and their splitting extrapolation for boundary integral equations of first kind on open arcsAppl. Numer. Math.2009592908292225608241175.6514210.1016/j.apnum.2009.06.006 RüdeUZhouAMulti-parameter extrapolation methods for boundary integral equationsAdv. Comput. Math.1988917319010.1023/A:1018989523557 ShiahYCTanCLBEM treatment of two-dimensional anisotropic field problems by direct domain mappingEng. Anal. Boundary Elem.19972034735110.1016/S0955-7997(97)00103-3 LiemCBLüTShihTMThe Splitting Extrapolation Method1995SingaporeWorld Scientific0875.65002 MeraNSElliottLInghamDBLesnicDA comparison of boundary element method formulations for steady state anisotropic heat conduction problemsEng. Anal. Boundary Elem.2001251151280982.8000910.1016/S0955-7997(00)00050-3 OkeyOOA boundary element-finite element equation solutions to flow in heterogeneous porous mediaTransp. Porous Media19983129331210.1023/A:1006529122626 HuangJLiZCChenILChengHDAdvanced quadrature methods and splitting extrapolation algorithms for first kind boundary integral equations of Laplace’s equation with discontinuity solutionsEng. Anal. Boundary Elem.2010341003100826803771244.6520110.1016/j.enganabound.2010.07.004 SidiAA new variable transformation for numerical integrationInt. Ser. Numer. Math.19931123593731248416 Jacob, B.: Dynamics of Fluids in Porous Media. American Elsevier Publishing Company, Inc (1972) ZhuRHuangJLüTMechanical quadrature methods and their splitting extrapolations for solving boundary integral equations of axisymmetric Laplace mixed boundary value problemsEng. Anal. Boundary Elem.20063039139810.1016/j.enganabound.2006.07.002 ChengC-ZZhouH-LHuZ-JNiuZ-RBoundary element analysis of the seepage flow field at interior points close to the dam fundationJ. Univ. Sci. Technol. China200636121308131305936403 DavisPMethods of Numerical Integration19842New YorkAcademic Press0537.65020 SidiAIsraeliMQuadrature methods for periodic singular and weakly singular Fredholm integral equationJ. Sci. Comput.198832012319817600662.6512210.1007/BF01061258 SloanIHSpenceAThe Galerkin method for integral equations of first-kind with logarithmic kernel: theoryIMA J. Numer. Anal.198881051229678460636.6514310.1093/imanum/8.1.105 HeX-MLüTSplitting extrapolation method for solving second-order parabolic equations with curved boundaries by using domain decomposition and d-quadratic isoparametric finite elementsInt. J. Comput. Math.20078476778123353671122.6508410.1080/00207160701458252 SaranenJThe modified quadrature method for logarithmic-kernel integral equations on closed curvesJ. Integral Equations Appl.1991357560011504080747.6510010.1216/jiea/1181075650 Georg, M., Hornberger, Jeffrey, P.R., Patricia, L.W., Keith, N.E.: Elements of Physical Hydrology. John Hopkins University Press (1998) CaoYHeX-MLüTA splitting extrapolation for solving nonlinear elliptic equations with d-quadratic finite elementsJ. Comput. Phys.200922810912224640701159.6509110.1016/j.jcp.2008.09.003 SvetlanaTRicardoCBoundary integral solutions of coupled Stokes and Darcy flowsJ. Comput. Phys.200922815817924640741188.7623210.1016/j.jcp.2008.09.011 YassineBSvetlanaTStokes–Darcy boundary integral solutions using preconditionersJ. Comput. Phys.20092288627864125587690563451110.1016/j.jcp.2009.08.014 QiLJLiaoHSSuPLLiSGThe technique rotationa lControlVolume in numerical simulation of anisotropic seepage flowJ. Sichuan Univ.200814011220 ElschnerJGrahamIGAn optimal order collocation method for first kind boundary integral equations on polygonsNumer. Math.19957013113206990827.6511710.1007/s002110050107 GrahamIQunLRui-FengXExtrapolation of Nystr öm solutions of boundary integral equations on non-smooth domainsJ. Comput. Math.19921023124411679250762.65062 HeX-MLüTA finite element splitting extrapolation for second order hyperbolic equationsSIAM J. Sci. Comput.2009314244426525665921205.6526910.1137/070703090 AtkinsonKThe Numerical Solution of Integral Equations of the Second Kind1997CambridgeCambridge University Press0899.6507710.1017/CBO9780511626340 ChenJTHongH-KChyuanSWBoundary element analysis and design in seepage problem using dual integral formulationFinite Elem. Anal. Des.1994171200825.7646810.1016/0168-874X(94)90017-5 SaranenJSloanIQuadrature methods for logarithmic-kernel integral equations on closed curvesIMA J. Numer. Anal.19921216718711645790751.6506910.1093/imanum/12.2.167 AtkinsonKESloanIHThe numerical solution of first-kind Logarithmic-kernel integral equation on smooth open arcsMath. Comp.199156193119139105208410.1090/S0025-5718-1991-1052084-0 LeopoldoPChristopherHFredericVOn a residual local projection method for the Darcy equationCompt. Rendus Math.200934717110511101171.7602610.1016/j.crma.2009.06.016 A Sidi (9563_CR32) 1993; 112 K Atkinson (9563_CR2) 1997 J Huang (9563_CR19) 2009; 59 J Huang (9563_CR18) 2010; 34 LJ Qi (9563_CR27) 2008; 140 PM Anselone (9563_CR1) 1971 P Davis (9563_CR11) 1984 J Huang (9563_CR17) 2009; 31 A Sidi (9563_CR33) 1988; 3 B Yassine (9563_CR39) 2009; 228 R Zhu (9563_CR40) 2006; 30 I Graham (9563_CR14) 1992; 10 T Svetlana (9563_CR36) 2009; 228 X-M He (9563_CR16) 2009; 31 A Mario (9563_CR23) 2003; 43 YC Shiah (9563_CR31) 1997; 20 IH Sloan (9563_CR34) 1988; 8 9563_CR20 GA Chandler (9563_CR7) 1984; 26 Y Yan (9563_CR38) 1988; 1 CB Liem (9563_CR22) 1995 EP Stephan (9563_CR35) 1984; 18 F Chatelin (9563_CR6) 1983 U Rüde (9563_CR28) 1988; 9 JT Chen (9563_CR8) 1994; 17 J Saranen (9563_CR29) 1991; 3 X-M He (9563_CR15) 2007; 84 OO Okey (9563_CR25) 1998; 31 NS Mera (9563_CR24) 2001; 25 Y Yan (9563_CR37) 1990; 54 Y Cao (9563_CR5) 2011; 235 C Pozrikidis (9563_CR26) 2003; 31 J Saranen (9563_CR30) 1992; 12 9563_CR13 Y Cao (9563_CR4) 2009; 228 C-Z Cheng (9563_CR9) 2006; 36 P Leopoldo (9563_CR21) 2009; 347 KE Atkinson (9563_CR3) 1991; 56 M Costabel (9563_CR10) 1988; 51 J Elschner (9563_CR12) 1995; 70 |
| References_xml | – reference: SidiAA new variable transformation for numerical integrationInt. Ser. Numer. Math.19931123593731248416 – reference: YanYThe collocation method for first-kind boundary integral equations on polygonal regionsMath. Comput.1990541391540685.6512110.1090/S0025-5718-1990-0995213-6 – reference: SvetlanaTRicardoCBoundary integral solutions of coupled Stokes and Darcy flowsJ. Comput. Phys.200922815817924640741188.7623210.1016/j.jcp.2008.09.011 – reference: ChenJTHongH-KChyuanSWBoundary element analysis and design in seepage problem using dual integral formulationFinite Elem. Anal. Des.1994171200825.7646810.1016/0168-874X(94)90017-5 – reference: Georg, M., Hornberger, Jeffrey, P.R., Patricia, L.W., Keith, N.E.: Elements of Physical Hydrology. John Hopkins University Press (1998) – reference: SloanIHSpenceAThe Galerkin method for integral equations of first-kind with logarithmic kernel: theoryIMA J. Numer. Anal.198881051229678460636.6514310.1093/imanum/8.1.105 – reference: RüdeUZhouAMulti-parameter extrapolation methods for boundary integral equationsAdv. Comput. Math.1988917319010.1023/A:1018989523557 – reference: ChengC-ZZhouH-LHuZ-JNiuZ-RBoundary element analysis of the seepage flow field at interior points close to the dam fundationJ. Univ. Sci. Technol. China200636121308131305936403 – reference: ZhuRHuangJLüTMechanical quadrature methods and their splitting extrapolations for solving boundary integral equations of axisymmetric Laplace mixed boundary value problemsEng. Anal. Boundary Elem.20063039139810.1016/j.enganabound.2006.07.002 – reference: CaoYHeXMLüTAn algorithm using the finite volume element method and its splitting extrapolationJ. Comput. Appl. Math.20112353734374227941661218.6511810.1016/j.cam.2011.01.019 – reference: HeX-MLüTA finite element splitting extrapolation for second order hyperbolic equationsSIAM J. Sci. Comput.2009314244426525665921205.6526910.1137/070703090 – reference: HuangJLüTLiZCThe mechanical quadrature methods and their splitting extrapolation for boundary integral equations of first kind on open arcsAppl. Numer. Math.2009592908292225608241175.6514210.1016/j.apnum.2009.06.006 – reference: StephanEPWendlandWLAn augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problemsAppl. Anal.19841818321976750010.1080/00036818408839520 – reference: AtkinsonKThe Numerical Solution of Integral Equations of the Second Kind1997CambridgeCambridge University Press0899.6507710.1017/CBO9780511626340 – reference: LeopoldoPChristopherHFredericVOn a residual local projection method for the Darcy equationCompt. Rendus Math.200934717110511101171.7602610.1016/j.crma.2009.06.016 – reference: PozrikidisCFarrowDAA model for fluid flow in solid tumorsAnn. Biomed. Eng.20033118119410.1114/1.1540103 – reference: QiLJLiaoHSSuPLLiSGThe technique rotationa lControlVolume in numerical simulation of anisotropic seepage flowJ. Sichuan Univ.200814011220 – reference: ShiahYCTanCLBEM treatment of two-dimensional anisotropic field problems by direct domain mappingEng. Anal. Boundary Elem.19972034735110.1016/S0955-7997(97)00103-3 – reference: ChatelinFSpectral Approximation of Linear Operator1983New YorkAcademic Press – reference: AnselonePMCollectively Compact Operator Approximation Theory1971Englewood CliffsPrentice-Hall0228.47001 – reference: LiemCBLüTShihTMThe Splitting Extrapolation Method1995SingaporeWorld Scientific0875.65002 – reference: MarioAGianmrcoMNull space algorithm and spanning trees in solving Darcy’s equationBIT Numerical Mathematics20034383984820588701046.6501710.1023/B:BITN.0000014568.20710.77 – reference: HuangJLiZCChenILChengHDAdvanced quadrature methods and splitting extrapolation algorithms for first kind boundary integral equations of Laplace’s equation with discontinuity solutionsEng. Anal. Boundary Elem.2010341003100826803771244.6520110.1016/j.enganabound.2010.07.004 – reference: YassineBSvetlanaTStokes–Darcy boundary integral solutions using preconditionersJ. Comput. 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| SubjectTerms | Adaptive algorithms Algebra Algorithms Asymptotic methods Boundary integral method Computer Science Convergence Extrapolation Integral equations Numeric Computing Numerical Analysis Numerical methods Original Paper Polygons Quadratures Splitting Theory of Computation |
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| Title | Splitting extrapolation algorithms for solving the boundary integral equations of anisotropic Darcy’s equation on polygons by mechanical quadrature methods |
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