# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions. (English) Zbl 1203.65225
Summary: We propose a tailored-finite-point method for a type of linear singular perturbation problem in two dimensions. Our finite point method has been tailored to some particular properties of the problem. Therefore, our new method can achieve very high accuracy with very coarse mesh even for very small $ϵ$, i.e. the boundary layers and interior layers do not need to be resolved numerically. In our numerical implementation, we study the classification of all the singular points for the corresponding degenerate first order linear dynamic system. We also study some cases with nonlinear coefficients. Our tailored finite point method is very efficient in both linear and nonlinear coefficients cases.
##### MSC:
 65N08 Finite volume methods (BVP of PDE)
##### References:
 [1] Berger, A.E., Han, H.D., Kellogg, R.B.: A priori estimates and analysis of a numerical method for a turning point problem. Math. Comput. 42, 465–492 (1984) · doi:10.1090/S0025-5718-1984-0736447-2 [2] Brayanov, I., Dimitrova, I.: Uniformly convergent high-order schemes for a 2D elliptic reaction-diffusion problem with anisotropic coefficients. Lect. Notes Comput. Sci. 2542, 395–402 (2003) [3] Cheng, M., Liu, G.R.: A novel finite point method for flow simulation. Int. J. Numer. Methods Fluids 39, 1161–1178 (2002) · Zbl 1053.76056 · doi:10.1002/fld.365 [4] Ge, L., Zhang, J.: High accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids. J. Comput. Phys. 171, 560–578 (2001) · Zbl 0990.65117 · doi:10.1006/jcph.2001.6794 [5] Han, H.D.: The artificial boundary method–numerical solutions of partial differential equations on unbounded domains. In: Li, T., Zhang, P. (eds.) Frontiers and Prospects of Contemporary Applied Mathematics. Higher Education Press/World Scientific, Singapore (2005) [6] Han, H., Huang, Z.: A tailored finite point method for the Helmholtz equation with high wave numbers in heterogeneous medium. J. Comput. Math. 26, 728–739 (2008) [7] Han, H., Huang, Z., Kellogg, B.: A Tailored finite point method for a singular perturbation problem on an unbounded domain. J. Sci. Comput. 36, 243–261 (2008) · Zbl 1203.65221 · doi:10.1007/s10915-008-9187-7 [8] Huang, Z.: Tailored finite point method for the interface problem. Netw. Heterogeneous Media 4, 91–106 (2009) · Zbl 1187.65128 · doi:10.3934/nhm.2009.4.91 [9] Hemker, P.W.: A singularly perturbed model problem for numerical computation. J. Comput. Appl. Math. 76, 277–285 (1996) · Zbl 0870.35020 · doi:10.1016/S0377-0427(96)00113-6 [10] Il’in, A.M.: Differencing scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes 6, 596–602 (1969) [11] Iooss, G., Joseph, D.D.: Elementary Stability and Bifurcation Theory. Springer, New York (1980) [12] Li, J., Chen, Y.: Uniform convergence analysis for singularly perturbed elliptic problems with parabolic layers. Numer. Math. Theor. Methods Appl. 1, 138–149 (2008) [13] Li, J., Navon, I.M.: Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems: convection-diffusion. Comput. Methods Appl. Mech. Eng. 162, 49–78 (1998) · Zbl 0936.65134 · doi:10.1016/S0045-7825(97)00329-0 [14] Lin, H., Atluri, S.N.: The Meshless Local Petrov-Galerkin (MLPG) method for solving incompressible Navier-Stokes equations. CMES 2, 117–142 (2001) [15] Mendeza, B., Velazquez, A.: Finite point solver for the simulation of 2-D laminar incompressible unsteady flows. Comput. Methods Appl. Mech. Eng. 193, 825–848 (2004) · Zbl 1106.76423 · doi:10.1016/j.cma.2003.11.010 [16] Miller, J.J.H.: On the convergence, uniformly in $ϵ$, of difference schemes for a two-point boundary singular perturbation problem. In: Hernker, P.W., Miller, J.J.H. (eds.) Numerical Analysis of Singular Perturbation Problems, pp. 467–474. Academic Press, San Diego (1979) [17] Morton, K.W.: Numerical Solution of Converction-Diffusion Problems. Applied Mathematics and Mathematical Computation, vol. 12. Chapman and Hall, London (1996) [18] Morton, K.W., Stynes, M., Süli, E.: Analysis of a cell-vertex finite volume method for convection-diffusion problems. Math. Comput. 66, 1389–1406 (1997) · Zbl 0885.65121 · doi:10.1090/S0025-5718-97-00886-7 [19] Oñate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L.: A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int. J. Numer. Methods Eng. 39, 3839–3866 (1996) · doi:10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R [20] Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, New York (1996) [21] Shishkin, G.I.: A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation. Numer. Math. Theor. Methods Appl. 1, 214–234 (2008) [22] Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005) · Zbl 1115.65108 · doi:10.1017/S0962492904000261 [23] Wesseling, P.: Uniform convergence of discretization error for a singular perturbation problem. Numer. Methods Partial Differ. Equ. 12, 657–671 (1996) · doi:10.1002/(SICI)1098-2426(199611)12:6<657::AID-NUM2>3.0.CO;2-R