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A high-order embedded domain method combining a predictor-corrector-Fourier-continuation-Gram method with an integral Fourier pseudospectral collocation method for solving linear partial differential equations in complex domains. (English) Zbl 1422.65426
Summary: Partial differential equations (PDEs) arise naturally in a wide variety of scientific areas and applications, and their numerical solutions are highly indispensable in many cases. Typical spectral/pseudospectral (PS) methods for solving PDEs work well only for regular domains such as rectangles or disks; however, the application of these methods to irregular domains is not straightforward and difficult enough to consider them less appealing as numerical tools. This research paper endeavors to take advantage of these methods in complex domains by introducing a novel, high-order numerical method that brings into play domain embedding into a regular, square computational domain in combination with integral reformulation, fully exponentially convergent Fourier PS collocation, and the Fourier-Continuation (FC)-Gram method integrated with a novel predictor-corrector-continuation algorithm to improve the accuracy of the extrapolated data. We developed some new formulas to construct the first- and second-order Fourier integration matrices (FIMs) based on equally-spaced nodes within the interval of integration. Modified FIMs were also developed to compute integral approximations of smooth, periodic functions when the upper limits of integration are any random points in the interval of integration. An algorithm for the fast and economic construction of the first-order FIM was also derived. A rounding error analysis based on numerical simulations demonstrates that the rounding errors in the calculation of the elements of the developed FIMs of size \(N\) are roughly of order less than or equal to \(O(N u_R)\), where \(u_R \approx 2 . 22 \times 1 0^{- 16}\) is the unit round-off of the double-precision floating-point system. The powerful features of the proposed method are illustrated through the study of the numerical solution of two-dimensional linear PDEs of Poisson type with constant coefficients and two different sets of boundary conditions. Two test problems are presented to demonstrate the overall capabilities of the proposed method. The current research paper investigations can be very useful when dealing with more complicated problems such as nonlinear PDEs in complex domains, or optimal control problems governed by linear/nonlinear PDEs in complex domains.
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N85 Fictitious domain methods for boundary value problems involving PDEs
Full Text: DOI
[1] Schwarz, H. A., Ueber einen Grenzübergang durch alternirendes Verfahren, (1870), Zürcher u. Furrer · JFM 02.0214.02
[2] Mathew, T., Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, vol. 61, (2008), Springer Science & Business Media
[3] Stupelis, L., (Navier-Stokes Equations in Irregular Domains. Navier-Stokes Equations in Irregular Domains, Mathematics and its Applications, (2013), Springer Netherlands) · Zbl 0837.35003
[4] Erhel, J.; Gander, M. J.; Halpern, L.; Pichot, G.; Sassi, T.; Widlund, O., Domain Decomposition Methods in Science and Engineering XXI, (2014), Springer · Zbl 1381.65002
[5] Glowinski, R.; Wheeler, M. F., Domain decomposition and mixed finite element methods for elliptic problems, (First International Symposium on Domain Decomposition Methods for Partial Differential Equations, (1988)), 144-172
[6] Gander, M. J.; Magoules, F.; Nataf, F., Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., 24, 1, 38-60, (2002) · Zbl 1021.65061
[7] P.-L. Lions, On the Schwarz alternating method. I, in: First International Symposium on Domain Decomposition Methods for Partial Differential Equations, Paris, France, 1988, pp. 1-42.
[8] Maday, Y.; Magoulès, F., Optimized Schwarz methods without overlap for highly heterogeneous media, Comput. Methods Appl. Mech. Engrg., 196, 8, 1541-1553, (2007) · Zbl 1173.74477
[9] Lions, P.-L., On the Schwarz alternating method. III: a variant for nonoverlapping subdomains, (Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, vol. 6, (1990), SIAM Philadelphia, PA), 202-223 · Zbl 0704.65090
[10] Martin, V., An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions, Appl. Numer. Math., 52, 4, 401-428, (2005) · Zbl 1070.65088
[11] Bouajaji, M. E.; Dolean, V.; Gander, M. J.; Lanteri, S., Optimized Schwarz methods for the time-harmonic Maxwell equations with damping, SIAM J. Sci. Comput., 34, 4, A2048-A2071, (2012) · Zbl 1259.78047
[12] Trefethen, L. N., Spectral Methods in MATLAB, (2000), SIAM: SIAM Philadelphia · Zbl 0953.68643
[13] Johnson, B. H., VAHM-A Vertically Averaged Hydrodynamic Model Using Boundary-Fitted coordinatesTechnical Report, (1980), DTIC Document
[14] Mai-Duy, N.; Tran-Cong, T., A Cartesian-grid collocation method based on radial-basis-function networks for solving PDEs in irregular domains, Numer. Methods Partial Differential Equations, 23, 5, 1192-1210, (2007) · Zbl 1129.65089
[15] Hirota, I.; Tokioka, T.; Nishiguchi, M., A direct solution of Poisson’s equation by generalized sweep-out method, J. Meteorol. Soc. Japan. II, 48, 2, 161-167, (1970)
[16] Brazier, P., An optimum SOR procedure for the solution of elliptic partial differential equations with any domain or coefficient set, Comput. Methods Appl. Mech. Engrg., 3, 3, 335-347, (1974) · Zbl 0278.65110
[17] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput. Struct., 11, 1-2, 83-95, (1980) · Zbl 0427.73077
[18] Chiu, C. P.; Wu, T. S., Study on the flow fields of irregular-shaped domains by an algebraic grid-generation technique, JSME Int. J. II, 34, 1, 69-77, (1991)
[19] Fogelson, A. L.; Keener, J. P., Immersed interface methods for Neumann and related problems in two and three dimensions, SIAM J. Sci. Comput., 22, 5, 1630-1654, (2001) · Zbl 0982.65112
[20] Ito, K.; Li, Z.; Kyei, Y., Higher-order, Cartesian grid based finite difference schemes for elliptic equations on irregular domains, SIAM J. Sci. Comput., 27, 1, 346-367, (2005) · Zbl 1087.65099
[21] Kozdon, J. E.; Dunham, E. M.; Nordström, J., Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods, J. Sci. Comput., 55, 1, 92-124, (2013) · Zbl 1278.86009
[22] Oberman, A. M.; Zwiers, I., Adaptive finite difference methods for nonlinear elliptic and parabolic partial differential equations with free boundaries, J. Sci. Comput., 68, 1, 231-251, (2016) · Zbl 1344.65101
[23] Towers, J. D., A source term method for Poisson problems on irregular domains, J. Comput. Phys., 361, 424-441, (2018) · Zbl 1422.65327
[24] Clough, R. W., The finite element method in plane stress analysis, (1960)
[25] Adini, A., Analysis of Shell Strutures by the Finite Element Method, (1961), University of California: University of California Berkeley
[26] Clough, R. W.; Wilson, E. L., Stress Analysis of a Gravity Dam by the Finite Element Method, (1962), Laboratório Nacional de Engenharia Civil
[27] Wilson, E., Finite Element Method in Two-Dimensional Problems, (1963), University of California, Berkeley, (D. Engg. thesis)
[28] Wilson, E. L.; Nickell, R. E., Application of the finite element method to heat conduction analysis, Nucl. Eng. Des., 4, 3, 276-286, (1966)
[29] Yamada, Y.; Yoshimura, N.; Sakurai, T., Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method, Int. J. Mech. Sci., 10, 5, 343-354, (1968) · Zbl 0159.56701
[30] Zlámal, M., On the finite element method, Numer. Math., 12, 5, 394-409, (1968) · Zbl 0176.16001
[31] J. Oden, Finite element applications in nonlinear structural analysis, in: Proceedings of the Conference on Finite Element Methods, Vanderbilt University, Tennessee, 1969, pp. 419-456.
[32] Bathe, K.-J., Finite Element Procedures, (2006), Klaus-Jurgen Bathe
[33] Szabo, B. A.; Babuška, I., Finite Element Analysis, (1991), John Wiley & Sons
[34] Sadiku, M. N., A simple introduction to finite element analysis of electromagnetic problems, IEEE Trans. Educ., 32, 2, 85-93, (1989)
[35] Lapidus, L.; Pinder, G., Numerical Solution of Partial Differential Equations in Science and Engineering, (2011), Wiley
[36] Li, H.; Liu, R., The discontinuous Galerkin finite element method for the 2D shallow water equations, Math. Comput. Simul., 56, 3, 223-233, (2001) · Zbl 0987.76054
[37] Cai, W.; Lee, H.; Oh, H.-S., Coupling of spectral methods and the p-version of the finite element method for elliptic boundary value problems containing singularities, J. Comput. Phys., 108, 2, 314-326, (1993) · Zbl 0790.65093
[38] Ferrigno, R.; Girault, H., Finite element simulation of electrochemical ac diffusional impedance. Application to recessed microdiscs, J. Electroanal. Soc., 492, 1, 1-6, (2000)
[39] Yang, Z.; Yuan, Z.; Nie, Y.; Wang, J.; Zhu, X.; Liu, F., Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains, J. Comput. Phys., 330, 863-883, (2017) · Zbl 1378.35330
[40] Song, P.; Wang, C.; Yotov, I., Domain decomposition for Stokes-Darcy flows with curved interfaces, Procedia Comput. Sci., 18, 1077-1086, (2013), 2013 International Conference on Computational Science
[41] Xue, Y.; Wang, C.; Liu, J.-G., Simple finite element numerical simulation of incompressible flow over non-rectangular domains and the super-convergence analysis, J. Sci. Comput., 65, 3, 1189-1216, (2015) · Zbl 1330.76081
[42] Garvie, M. R.; Burkardt, J.; Morgan, J., Simple finite element methods for approximating Predator-Prey dynamics in two dimensions using Matlab, Bull. Math. Biol., 77, 3, 548-578, (2015) · Zbl 1332.92053
[43] Pozrikidis, C., Introduction to Finite and Spectral Element Methods Using MATLAB, (2014), CRC Press · Zbl 1337.65001
[44] Ding, H.; Shu, C.; Yeo, K.; Xu, D., Development of least-square-based two-dimensional finite-difference schemes and their application to simulate natural convection in a cavity, Comput. & Fluids, 33, 1, 137-154, (2004) · Zbl 1033.76039
[45] Mirzadeh, M.; Theillard, M.; Gibou, F., A second-order discretization of the nonlinear Poisson-Boltzmann equation over irregular geometries using non-graded adaptive Cartesian grids, J. Comput. Phys., 230, 5, 2125-2140, (2011) · Zbl 1390.82056
[46] P. Tota, Z. Wang, Meshfree Euler solver using local radial basis functions for inviscid compressible flows, in: 18th AIAA Computational Fluid Dynamics Conference, 2007, p. 4581.
[47] Eigel, M.; George, E.; Kirkilionis, M., The partition of unity meshfree method for solving transport-reaction equations on complex domains: implementation and applications in the life sciences, (Meshfree Methods for Partial Differential Equations IV, (2008), Springer), 69-93 · Zbl 1156.92012
[48] Kong, W.; Wu, X., Chebyshev tau matrix method for Poisson-type equations in irregular domain, J. Comput. Appl. Math., 228, 1, 158-167, (2009) · Zbl 1165.65081
[49] Khoshfetrat, A.; Abedini, M., A hybrid DQ/LMQRBF-DQ approach for numerical solution of Poisson-type and Burger’s equations in irregular domain, Appl. Math. Model., 36, 5, 1885-1901, (2012) · Zbl 1243.65147
[50] Singh, S.; Bhargava, R., Numerical study of natural convection within a wavy enclosure using meshfree approach: Effect of corner heating, Sci. World J., 2014, (2014)
[51] Siraj-ul-Islam; Ahmad, I., Local meshless method for PDEs arising from models of wound healing, Appl. Math. Model., 48, 688-710, (2017)
[52] Rostamy, D.; Emamjome, M.; Abbasbandy, S., A meshless technique based on the pseudospectral radial basis functions method for solving the two-dimensional hyperbolic telegraph equation, Eur. Phys. J. Plus, 132, 6, 263, (2017)
[53] Madsen, N. K.; Ziolkowski, R. W., Numerical solution of Maxwell’s equations in the time domain using irregular nonorthogonal grids, Wave Motion, 10, 6, 583-596, (1988) · Zbl 0672.73029
[54] Wu, Y.; Shu, C.; Qiu, J.; Tani, J., Implementation of multi-grid approach in domain-free discretization method to speed up convergence, Comput. Methods Appl. Mech. Engrg., 192, 20, 2425-2438, (2003) · Zbl 1034.76048
[55] McFall, K. S.; Mahan, J. R., Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions, IEEE Trans. Neural Netw., 20, 8, 1221-1233, (2009)
[56] Zhi, S.; Yan-Hua, X.; Jun-Ping, Z., Haar wavelets method for solving Poisson equations with jump conditions in irregular domain, Adv. Comput. Math., 42, 4, 995-1012, (2016) · Zbl 1348.35072
[57] Ghimire, B. K.; Tian, H.; Lamichhane, A., Numerical solutions of elliptic partial differential equations using Chebyshev polynomials, Comput. Math. Appl., 72, 4, 1042-1054, (2016) · Zbl 1359.65275
[58] Hosseinverdi, S.; Fasel, H. F., An efficient, high-order method for solving Poisson equation for immersed boundaries: Combination of compact difference and multiscale multigrid methods, J. Comput. Phys., 374, 912-940, (2018)
[59] Liu, G.; Ma, W.; Ma, H.; Zhu, L., A multiple-scale higher order polynomial collocation method for 2D and 3D elliptic partial differential equations with variable coefficients, Appl. Math. Comput., 331, 430-444, (2018)
[60] Bochkov, D.; Gibou, F., Solving Poisson-type equations with robin boundary conditions on piecewise smooth interfaces, J. Comput. Phys., 376, 1156-1198, (2019)
[61] Orszag, S. A., Numerical methods for the simulation of turbulence, Phys. Fluids, 12, 12, II-250, (1969) · Zbl 0217.25803
[62] Orszag, S. A., Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50, 4, 659-703, (1971) · Zbl 0237.76027
[63] Patterson, G.; Orszag, S. A., Spectral calculations of isotropic turbulence: Efficient removal of aliasing interactions, Phys. Fluids, 14, 11, 2538-2541, (1971) · Zbl 0225.76033
[64] Kreiss, H.-O.; Oliger, J., Comparison of accurate methods for the integration of hyperbolic equations, Tellus, 24, 3, 199-215, (1972)
[65] Orszag, S. A., Spectral methods for problems in complex geometries, J. Comput. Phys., 37, 1, 70-92, (1980) · Zbl 0476.65078
[66] Mohd-Yusof, J., Methods for complex geometries, Ann. Res. Briefs-1998, 325, (1998)
[67] Shen, J.; Tang, T.; Wang, L.-L., Spectral Methods: Algorithms, Analysis and Applications, vol. 41, (2011), Springer Science & Business Media
[68] Elgindy, K. T.; Smith-Miles, K. A., Fast, accurate, and small-scale direct trajectory optimization using a Gegenbauer transcription method, J. Comput. Appl. Math., 251, 93-116, (2013) · Zbl 1290.49055
[69] Behroozifar, M., Spectral method for solving high order nonlinear boundary value problems via operational matrices, BIT Numerical Mathematics, 55, 4, 901-925, (2015) · Zbl 1332.65113
[70] Quarteroni, A.; Valli, A., (Numerical Approximation of Partial Differential Equations. Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, vol. 23, (1994), Springer-Verlag), 2nd corr. printing · Zbl 0803.65088
[71] Mason, J.; Handscomb, D., Chebyshev Polynomials, (2002), CRC Press · Zbl 1015.33001
[72] Zayernouri, M.; Ainsworth, M.; Karniadakis, G., A unified Petrov-Galerkin spectral method for fractional PDEs, Comput. Methods Appl. Mech. Engrg., 283, 1545-1569, (2015) · Zbl 1425.65127
[73] Luo, X.-H.; Li, B.-W.; Zhang, J.-K.; Hu, Z.-M., Simulation of thermal radiation effects on MHD free convection in a square cavity using the Chebyshev collocation spectral method, Numer. Heat Transfer A, 66, 7, 792-815, (2014)
[74] Zhou, J.; Yang, D., Legendre-Galerkin spectral methods for optimal control problems with integral constraint for state in one dimension, Comput. Optim. Appl., (2014)
[75] Graef, J.; Kong, L.; Wang, M., A Chebyshev spectral method for solving Riemann-Liouville fractional boundary value problems, Appl. Math. Comput., 241, 140-150, (2014) · Zbl 1337.65104
[76] Zhuang, Q.; Ren, Q., Numerical approximation of a nonlinear fourth-order integro-differential equation by spectral method, Appl. Math. Comput., 232, 775-783, (2014) · Zbl 1410.65512
[77] Brand, S.; Tildesley, M.; Keeling, M., Rapid simulation of spatial epidemics: A spectral method, J. Theoret. Biol., 370, 121-134, (2015) · Zbl 1337.92203
[78] Feng, R.; Volkmer, H., Spectral methods for the calculation of risk measures for variable annuity guaranteed benefits, Astin Bull., 44, 3, 653-681, (2014)
[79] Ehrendorfer, M., Spectral Numerical Weather Prediction Models, vol. 124, (2012), SIAM · Zbl 1239.86001
[80] Elgindy, K. T.; Hedar, A., A new robust line search technique based on Chebyshev polynomials, Appl. Math. Comput., 206, 2, 853-866, (2008) · Zbl 1163.65038
[81] Kang, W.; Bedrossian, N., Pseudospectral optimal control theory makes debut flight, saves NASA \(1M in under three hours, SIAM News, 40, 7, (2007\)
[82] Benson, D. A.; Huntington, G. T.; Thorvaldsen, T. P.; Rao, A. V., Direct trajectory optimization and costate estimation via an orthogonal collocation method, J. Guid. Control Dyn., 29, 6, 1435-1440, (2006)
[83] Elgindy, K. T., Generation of higher order pseudospectral integration matrices, Appl. Math. Comput., 209, 2, 153-161, (2009) · Zbl 1170.65105
[84] Gong, Q.; Kang, W.; Ross, I., A pseudospectral method for the optimal control of constrained feedback linearizable systems, IEEE Trans. Automat. Control, 51, 7, 1115-1129, (2006) · Zbl 1366.49035
[85] Vera, S.; Cobano, J. A.; Heredia, G.; Ollero, A., An hp-adaptative pseudospectral method for conflict resolution in converging air traffic, (Moreira, A. P.; Matos, A.; Veiga, G., CONTROLO’2014-Proceedings of the 11th Portuguese Conference on Automatic Control. CONTROLO’2014-Proceedings of the 11th Portuguese Conference on Automatic Control, Lecture Notes in Electrical Engineering, vol. 321, (2015), Springer International Publishing), 333-343
[86] Aly, E.; Vajravelu, K., Exact and numerical solutions of MHD nano boundary-layer flows over stretching surfaces in a porous medium, Appl. Math. Comput., 232, 191-204, (2014) · Zbl 1410.76470
[87] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics, (1988), Springer-Verlag: Springer-Verlag Berlin · Zbl 0658.76001
[88] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., (Spectral Methods: Fundamentals in Single Domains. Spectral Methods: Fundamentals in Single Domains, Scientific Computation, (2006), Springer: Springer Berlin; NY) · Zbl 1093.76002
[89] Boyd, J. P., Chebyshev and Fourier Spectral Methods, (2001), Courier Corporation · Zbl 0994.65128
[90] Guo, B. Y., Spectral Methods and their Applications, (1998), World Scientific: World Scientific Singapore
[91] Hesthaven, J. S.; Gottlieb, S.; Gottlieb, D., (Spectral Methods for Time-Dependent Problems. Spectral Methods for Time-Dependent Problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21, (2007), Cambridge University Press) · Zbl 1111.65093
[92] Orszag, S. A.; Gottlieb, D., High resolution spectral calculations of inviscid compressible flows, (Approximation Methods for Navier-Stokes Problems, (1980), Springer), 381-398
[93] Doha, E.; Bhrawy, A.; Hafez, R., On shifted Jacobi spectral method for high-order multi-point boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 17, 10, 3802-3810, (2012) · Zbl 1251.65112
[94] Fornberg, B., (A Practical Guide to Pseudospectral Methods. A Practical Guide to Pseudospectral Methods, Cambridge Monographs on Applied and Computational Mathematics, vol. 1, (1996), Cambridge University Press, Cambridge) · Zbl 0844.65084
[95] Livermore, P. W., The spherical harmonic spectrum of a function with algebraic singularities, J. Fourier Anal. Appl., 18, 6, 1146-1166, (2012) · Zbl 1262.33012
[96] Elgindy, K. T.; Smith-Miles, K. A., Optimal Gegenbauer quadrature over arbitrary integration nodes, J. Comput. Appl. Math., 242, 82-106, (2013) · Zbl 1255.65071
[97] Elgindy, K., Gegenbauer Collocation Integration Methods: Advances in Computational Optimal Control Theory, (2013), School of Mathematical Sciences, Faculty of Science, Monash University: School of Mathematical Sciences, Faculty of Science, Monash University Australia-Victoria, (Ph.D thesis)
[98] Tapia, J. J.; López, P. G., Adaptive pseudospectral solution of a diffuse interface model, J. Comput. Appl. Math., 224, 1, 101-117, (2009) · Zbl 1394.76082
[99] Baltensperger, R.; Trummer, M. R., Spectral differencing with a twist, SIAM J. Sci. Comput., 24, 5, 1465-1487, (2003) · Zbl 1034.65016
[100] Bayliss, A.; Class, A.; Matkowsky, B. J., Roundoff error in computing derivatives using the Chebyshev differentiation matrix, J. Comput. Phys., 116, 2, 380-383, (1995) · Zbl 0826.65014
[101] Don, W. S.; Solomonoff, A., Accuracy and speed in computing the Chebyshev collocation derivative, SIAM J. Sci. Comput., 16, 6, 1253-1268, (1995) · Zbl 0840.65010
[102] Don, W. S.; Solomonoff, A., Accuracy enhancement for higher derivatives using Chebyshev collocation and a mapping technique, SIAM J. Sci. Comput., 18, 4, 1040-1055, (1997) · Zbl 0906.65019
[103] Roger, P., Spectral Methods for Incompressible Viscous Flow, (2002), Springer, New York · Zbl 1005.76001
[104] Tang, T.; Trummer, M. R., Boundary layer resolving pseudospectral methods for singular perturbation problems, SIAM J. Sci. Comput., 17, 2, 430-438, (1996) · Zbl 0851.65058
[105] Tong, C. H.; Chan, T. F.; Kuo, C. C.J., A domain decomposition preconditioner based on a change to a multilevel nodal basis, SIAM J. Sci. Stat. Comput., 12, 6, 1486-1495, (1991) · Zbl 0744.65084
[106] Mandel, J., Two-level domain decomposition preconditioning for the p-version finite element method in three dimensions, Internat. J. Numer. Methods Engrg., 29, 5, 1095-1108, (1990) · Zbl 0712.73091
[107] Tsuji, P.; Poulson, J.; Engquist, B.; Ying, L., Sweeping preconditioners for elastic wave propagation with spectral element methods, ESAIM Math. Model. Numer. Anal., 48, 02, 433-447, (2014) · Zbl 1423.74950
[108] Smith, B.; Bjorstad, P.; Gropp, W., Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, (2004), Cambridge university press
[109] Morchoisne, Y., Résolution des équations de Navier-Stokes par une méthode spectrale de sous-domaines, C.-R. du, 3, (1983)
[110] Patera, A. T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys., 54, 3, 468-488, (1984) · Zbl 0535.76035
[111] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, (2007), Springer-Verlag: Springer-Verlag Berlin; New York · Zbl 1121.76001
[112] Farhat, C.; Maman, N.; Brown, G. W., Mesh partitioning for implicit computations via iterative domain decomposition: impact and optimization of the subdomain aspect ratio, Internat. J. Numer. Methods Engrg., 38, 6, 989-1000, (1995) · Zbl 0825.73780
[113] Margetts, L., Parallel Finite Element Analysis, (2002), University of Manchester
[114] Henderson, R.; Karniadakis, G. E., Hybrid spectral-element-low-order methods for incompressible flows, J. Sci. Comput., 6, 2, 79-100, (1991) · Zbl 0741.76058
[115] Lou, Z.; Jin, J.-M., A novel dual-field time-domain finite-element domain-decomposition method for computational electromagnetics, IEEE Trans. Antennas Propag., 54, 6, 1850-1862, (2006) · Zbl 1369.78684
[116] Giraldo, F. X.; Hesthaven, J. S.; Warburton, T., Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations, J. Comput. Phys., 181, 2, 499-525, (2002) · Zbl 1178.76268
[117] Bueno-Orovio, A., Fourier embedded domain methods: periodic and C\({}^\infty\) extension of a function defined on an irregular region to a rectangle via convolution with Gaussian kernels, Appl. Math. Comput., 183, 2, 813-818, (2006) · Zbl 1105.65354
[118] Hyman, M. A., Non-iterative numerical solution of boundary-value problems, Appl. Sci. Res. B, 2, 1, 325-351, (1952) · Zbl 0048.10201
[119] Saul’ev, V. K., On the solution of some boundary value problems on high performance computers by fictitious domain method, Siberian Math. J., 4, 4, 912-925, (1963)
[120] Badea, L.; Daripa, P., On a Fourier method of embedding domains using an optimal distributed control, Numer. Algorithms, 32, 2-4, 261-273, (2003) · Zbl 1079.65124
[121] Badea, L.; Daripa, P., A domain embedding method using the optimal distributed control and a fast algorithm, Numer. Algorithms, 36, 2, 95-112, (2004) · Zbl 1052.93021
[122] Zhang, S., A domain embedding method for mixed boundary value problems, C. R. Math., 343, 4, 287-290, (2006) · Zbl 1100.65106
[123] Lui, S., Spectral domain embedding for elliptic PDEs in complex domains, J. Comput. Appl. Math., 225, 2, 541-557, (2009) · Zbl 1160.65346
[124] Zhu, J.; Ma, Y., Fictitious domain method with penalty for an incompressible fluid, Numer. Methods Partial Differential Equations, 26, 1, 229-238, (2010) · Zbl 1425.65128
[125] Zhong, Y., The Spectral Domain Embedding Method for Partial Differential Equations on Irregular Domains, (2013), Simon Fraser University, (Ph.D thesis)
[126] Falletta, S.; Monegato, G., A Fictitious Domain Approach for Wave Propagation Problems in Unbounded Domains, (2015), Institute of Structural Analysis and Antiseismic Research School of Civil Engineering National Technical University of Athens (NTUA)
[127] Boyd, J. P., Fourier embedded domain methods: extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and C\({}^\infty\), Appl. Math. Comput., 161, 2, 591-597, (2005) · Zbl 1061.65127
[128] Chen, J.-H., A domain-extension radial basis function collocation method for heat transfer in irregular domains, J. Chin. Inst. Eng., 32, 3, 319-326, (2009)
[129] Wang, Z.-Q.; Li, S.; Ping, Y.; Jiang, J.; Ma, T.-F., A highly accurate regular domain collocation method for solving potential problems in the irregular doubly connected domains, Math. Probl. Eng., 2014, (2014) · Zbl 1407.65313
[130] Mayo, A., The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21, 2, 285-299, (1984) · Zbl 1131.65303
[131] Jomaa, Z.; Macaskill, C., The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions, J. Comput. Phys., 202, 2, 488-506, (2005) · Zbl 1061.65107
[132] Bueno-Orovio, A.; Pérez-García, V. M.; Fenton, F. H., Spectral methods for partial differential equations in irregular domains: the spectral smoothed boundary method, SIAM J. Sci. Comput., 28, 3, 886-900, (2006) · Zbl 1114.65119
[133] Stein, D. B.; Guy, R. D.; Thomases, B., Immersed boundary smooth extension: a high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods, J. Comput. Phys., 304, 252-274, (2016) · Zbl 1349.65656
[134] Elghaoui, M.; Pasquetti, R., A spectral embedding method applied to the advection-diffusion equation, J. Comput. Phys., 125, 2, 464-476, (1996) · Zbl 0852.65086
[135] Atamian, C.; Dinh, Q.; Glowinski, R.; He, J.; Périaux, J., Control approach to fictitious-domain methods. Application to fluid dynamics and electro-magnetics, (Glowinski, R.; Kuznetsov, Y.; Meurant, G.; Périaux, J.; Widlund, O. B., Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, (1991), SIAM: SIAM Philadelphia), 275-309 · Zbl 0768.76042
[136] Bruno, O. P.; Lyon, M., High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements, J. Comput. Phys., 229, 6, 2009-2033, (2010) · Zbl 1185.65184
[137] Boyd, J. P., A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds, J. Comput. Phys., 178, 1, 118-160, (2002) · Zbl 0999.65132
[138] Bruno, O. P., Fast, high-order, high-frequency integral methods for computational acoustics and electromagnetics, (Topics in Computational Wave Propagation, (2003), Springer), 43-82 · Zbl 1051.78009
[139] Bruno, O. P.; Han, Y.; Pohlman, M. M., Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis, J. Comput. Phys., 227, 2, 1094-1125, (2007) · Zbl 1128.65017
[140] Lyon, M., A fast algorithm for Fourier continuation, SIAM J. Sci. Comput., 33, 6, 3241-3260, (2011) · Zbl 1255.65253
[141] Matthysen, R.; Huybrechs, D., Fast algorithms for the computation of Fourier extensions of arbitrary length, SIAM J. Sci. Comput., 38, 2, A899-A922, (2016) · Zbl 1337.65181
[142] Françolin, C. C.; Benson, D. A.; Hager, W. W.; Rao, A. V., Costate approximation in optimal control using integral Gaussian quadrature orthogonal collocation methods, Optim. Control Appl. Methods, (2014) · Zbl 1329.49045
[143] Tang, X., Efficient and stable generation of higher-order pseudospectral integration matrices, Appl. Math. Comput., 261, 60-67, (2015) · Zbl 1410.65506
[144] Coutsias, E. A.; Hagstrom, T.; Torres, D., An efficient spectral method for ordinary differential equations with rational function coefficients, Math. Comp., 65, 214, 611-635, (1996) · Zbl 0846.65037
[145] Greengard, L., Spectral integration and two-point boundary value problems, SIAM J. Numer. Anal., 28, 4, 1071-1080, (1991) · Zbl 0731.65064
[146] Viswanath, D., Spectral integration of linear boundary value problems, J. Comput. Appl. Math., 290, 159-173, (2015) · Zbl 1330.65035
[147] Driscoll, T. A., Automatic spectral collocation for integral, integro-differential, and integrally reformulated differential equations, J. Comput. Phys., 229, 17, 5980-5998, (2010) · Zbl 1195.65225
[148] Olver, S.; Townsend, A., A fast and well-conditioned spectral method, SIAM Rev., 55, 3, 462-489, (2013) · Zbl 1273.65182
[149] El-Gendi, S. E., Chebyshev solution of differential, integral, and integro-differential equations, Comput. J., 12, 3, 282-287, (1969) · Zbl 0198.50201
[150] Elgindy, K. T., Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted Gegenbauer integral pseudospectral method, J. Ind. Manag. Optim., 14, 2, 473-496, (2018) · Zbl 1412.65014
[151] Elgindy, K. T.; Refat, H. M., High-order shifted Gegenbauer integral pseudo-spectral method for solving differential equations of Lane-Emden type, Appl. Numer. Math., 128, 98-124, (2018) · Zbl 1415.65174
[152] Elgindy, K. T.; Dahy, S. A., High-order numerical solution of viscous Burgers’ equation using a Cole-Hopf barycentric Gegenbauer integral pseudospectral method, Math. Methods Appl. Sci., 41, 16, 6226-6251, (2018) · Zbl 1407.65212
[153] Elgindy, K. T.; Karasözen, B., High-order integral nodal discontinuous Gegenbauer-Galerkin method for solving viscous Burgers’ equation, Int. J. Comput. Math., just-accepted, 1-44, (2018)
[154] Elgindy, K. T., High-order numerical solution of second-order one-dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method, Numer. Methods Partial Differential Equations, 32, 1, 307-349, (2016) · Zbl 1346.65052
[155] K.T. Elgindy, B. Karasözen, Distributed optimal control of viscous Burgers’ equation via a high-order, linearization, integral, nodal discontinuous Gegenbauer-Galerkin method, 2018, submitted for publication.
[156] Elgindy, K. T., High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures, Appl. Numer. Math., 113, 1-25, (2017) · Zbl 1416.65230
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