An exponential convergence approximation to singularly perturbed problems by log orthogonal functions. (English) Zbl 1492.65339
Summary: A new spectral Galerkin method based on Log orthogonal functions (LOFs) is proposed for singularly perturbed problems (SSPs). The method uses only (orthogonal) basis functions to approximate the boundary layer function \(\exp(-x/\varepsilon)\) without help from any specially designed mesh. An error estimate for the boundary layer function is derived and shows the exponential convergence of the new method. Ample numerical examples in 1d and 2d validate the high-efficiency and robustness of the Log spectral/spectral-element method for SSPs.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 41A30 | Approximation by other special function classes |
| 34B08 | Parameter dependent boundary value problems for ordinary differential equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
log orthogonal functions; mapped spectral methods; singular perturbed problem; exponential convergence; convection-diffusion equationReferences:
| [1] | Adjerid, S.; Aiffa, M.; Flaherty, JE, High-order finite element methods for singularly perturbed elliptic and parabolic problems, SIAM J. Appl. Math., 55, 2, 520-543 (1995) · Zbl 0827.65097 · doi:10.1137/S0036139993269345 |
| [2] | Babuska, I., Aziz, A.K.: Lectures on mathematical foundations of the finite element method. Maryland Univ. College Park Inst. for Fluid Dynamics and Applied Mathematics, Technical report (1972) |
| [3] | Bernardi, C.; Maday, Y., Spectral Methods. Handbook of Numerical Analysis, 5, 209-485 (1997) · doi:10.1016/S1570-8659(97)80003-8 |
| [4] | Boyd, J.P., Marilyn, T., Eliot, P.T.S.: Chebyshev and fourier spectral methods. Dover Publications (2000) |
| [5] | Brenner, S., Scott, R.: The mathematical theory of finite element methods, volume 15. Springer Science & Business Media (2007) |
| [6] | Canuto, C.G., Hussaini, M.Y., Quarteroni, A.M., Zang, T.A.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation) (2007) · Zbl 1121.76001 |
| [7] | Chen, YP, Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution, J. Comput. Appl. Math., 159, 1, 25-34 (2003) · Zbl 1044.65066 · doi:10.1016/S0377-0427(03)00563-6 |
| [8] | Chen, S., High-accuracy numerical approximations to several singularly perturbed problems and singular integral equations by enriched spectral Galerkin methods, J. Math. Study, 53, 2, 143-158 (2020) · Zbl 1463.65393 · doi:10.4208/jms.v53n2.20.02 |
| [9] | Chen, S., Shen, J.: Log orthogonal functions: their approximation properties and applications to fractional differential equations. IMA J. Numer. Anal. doi:10.1093/imanum/draa087 |
| [10] | Chen, S.; Shen, J.; Zhang, ZM; Zhou, Z., A spectrally accurate approximation to subdiffusion equations using the log orthogonal functions, SIAM J. Sci. Comput., 42, 2, A849-A877 (2020) · Zbl 1434.65291 · doi:10.1137/19M1281927 |
| [11] | Ciarlet, PG, The finite element method for elliptic problems, Class. Appl. Math., 40, 1-511 (2002) · Zbl 0999.65129 |
| [12] | Friedrichs, K.O., Wasow, W.R.: Singular perturbations of non-linear oscillations. Duke Math. J. 13(3) (1946) · Zbl 0061.19509 |
| [13] | Gie, G.M., Kelliher, J.P., Mazzucato, A.L.: Boundary layers for the Navier-Stokes equations linearized around a stationary Euler flow. J. Math. Fluid Mech. (2018) · Zbl 1404.35016 |
| [14] | Gottlieb, D., Orszag, S.A.: Numerical analysis of spectral methods: theory and applications, volume 26. SIAM (1977) · Zbl 0412.65058 |
| [15] | Guo, B.Y.: Spectral methods and their applications. World Scientific (1998) · Zbl 0906.65110 |
| [16] | Jung, CY; Temam, R., Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point, J. Math. Phys., 48, 6, 065301-065301 (2007) · Zbl 1144.81363 · doi:10.1063/1.2347899 |
| [17] | Kopteva, N.; O’ Riordan, E., Shishkin meshes in the numerical solution of singularly perturbed differential equations, Int. J. Numer. Anal. Model., 7, 3, 393-415 (2010) · Zbl 1197.65094 |
| [18] | Kumar, M.; Singh, P.; Mishra, HK, A recent survey on computational techniques for solving singularly perturbed boundary value problems, Int. J. Comput. Math., 84, 10, 1439-1463 (2007) · Zbl 1127.65053 · doi:10.1080/00207160701295712 |
| [19] | Levinson, N.: The first boundary value problem for \(\varepsilon \delta u+a(x,y)u_x+b(x,y)u_y+c(x,y)u=d(x,y)\) for small \(varepsilon\). Ann. Math, 428-445 (1950) · Zbl 0036.06801 |
| [20] | Linss, T.: Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem. IMA J. Numer. Anal. (2000) · Zbl 0966.65083 |
| [21] | Liu, WB; Shen, J., A new efficient spectral Galerkin method for singular perturbation problems, J. Sci. Comput., 11, 4, 411-437 (1996) · Zbl 0891.76066 · doi:10.1007/BF02088955 |
| [22] | Liu, WB; Tang, T., Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems, Appl. Numer. Math., 38, 3, 315-345 (2001) · Zbl 1023.65115 · doi:10.1016/S0168-9274(01)00036-8 |
| [23] | Maclachlan, S., Madden, N.: Robust solution of singularly perturbed problems using multigrid methods. SIAM J. Sci. Comput. 35(5) (2013) · Zbl 1290.65097 |
| [24] | Melenk, JM, On the robust exponential convergence of hp finite element methods for problems with boundary layers, IMA J. Numer. Anal., 4, 577-601 (1996) · Zbl 0887.65106 |
| [25] | Melenk, J.M.: hp-finite element methods for singular perturbations (2002) · Zbl 1021.65055 |
| [26] | Miller, JJH; O’Riordan, E.; Shishkin, GI, Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions, SIAM J. Numer. Anal., 18, 2, 262-276 (1981) · Zbl 0457.65064 · doi:10.1137/0718019 |
| [27] | Oleini, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory. Chapman Hall /CRC, (1999) · Zbl 0928.76002 |
| [28] | O’Malley, J., Robert E.: Featured review: robust numerical methods for singularly perturbed differential equations. Springer (2008) |
| [29] | O’Malley; Robert, E., Singularly perturbed linear two-point boundary value problems, SIAM Rev., 50, 3, 459-482 (2008) · Zbl 1362.34092 · doi:10.1137/060662058 |
| [30] | Pearson, CE, On a differential equation of boundary layer type, Studi. Appl. Math., 47, 1-4, 134-154 (1968) · Zbl 0167.15801 |
| [31] | Reilly, MJO; O’Riordan, E., A shishkin mesh for a singularly perturbed riccati equation, J. Comput. Appl. Math., 182, 2, 372-387 (2005) · Zbl 1079.65088 · doi:10.1016/j.cam.2004.12.018 |
| [32] | Roos, HG, Robust numerical methods for singularly perturbed differential equations: a survey covering 2008-2012, Isrn Appl. Math., 223-252, 2015 (2012) · Zbl 1264.65116 |
| [33] | Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems (Second edition). Springer Series in Computational Mathematics. Springer-Verlag (2008) · Zbl 1155.65087 |
| [34] | Schwab, C.; Suri, M., The p and hp versions of the finite element method for problems with boundary layers, Math. Computat., 65, 216, 1403-1429 (1996) · Zbl 0853.65115 · doi:10.1090/S0025-5718-96-00781-8 |
| [35] | Schwab, C., Suri, M., Xenophontos.: The hp finite element method for problems in mechanics with boundary layer. Comput. Methods . Appl. Mech. Engineering (1998) · Zbl 0959.74073 |
| [36] | Shen, J., Tang, T., Wang, L.L.: Spectral methods: algorithms, analysis and applications. Springer (2011) · Zbl 1227.65117 |
| [37] | Shishkin, G.I., Shishkina L.P.: Difference Methods for Singular Perturbation Problems. CHAPMAN HALL/CRC (2009) · Zbl 1163.65062 |
| [38] | Stynes, M., and Stynes, D.: Convection-diffusion problems: an introduction to their analysis and numerical solution. Graduate Studies in Mathematics Vol. 196, American Math. Society, Providence, (2018) · Zbl 1422.65005 |
| [39] | Suri, M.: The p and hp finite element method for problems on thin domains. Elsevier Science Publishers B. V. (2001) · Zbl 0974.65102 |
| [40] | Tang, T., Trummer, M.R.: Boundary layer resolving pseudospectral methods for singular perturbation problems. SIAM J. Sci. Comput. 17(2) (1996) · Zbl 0851.65058 |
| [41] | Temam, R.; Wang, X., Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case, J. Differ. Equ., 179, 2, 647-686 (2002) · Zbl 0997.35042 · doi:10.1006/jdeq.2001.4038 |
| [42] | Temam, R., Petcu, M., Jung, C. Y.: Singular perturbation analysis on a homogeneous ocean circulation model. Anal. Appl. (2011) · Zbl 1229.76091 |
| [43] | Wang, LL; Shen, J., Error analysis for mapped jacobi spectral methods, J. Sci. Comput., 24, 2, 183-218 (2005) · Zbl 1082.65091 · doi:10.1007/s10915-004-4613-y |
| [44] | Wu, YJ; Na, Z.; Yuan, JY, A robust adaptive method for singularly perturbed convection-diffusion problem with two small parameters, Comput. Math. Appl., 66, 6, 996-1009 (2013) · Zbl 07863335 · doi:10.1016/j.camwa.2013.06.025 |
| [45] | Zhang, ZM, Finite element superconvergence on shishkin mesh for 2-d convection-diffusion problems, Math. Comput., 72, 243, 1147-1177 (2003) · Zbl 1019.65091 · doi:10.1090/S0025-5718-03-01486-8 |
| [46] | Zhang, ZM, Finite element superconvergence approximation for one-dimensional singularly perturbed problems (p), Numer. Methods Part. Differ. Equ., 18, 3, 374-395 (2010) · Zbl 1002.65088 · doi:10.1002/num.10001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
