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Stability and accuracy of adapted finite element methods for singularly perturbed problems. (English) Zbl 1146.65059
The authors study the stability and accuracy of the standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) for the following one-dimensional linear singularly perturbed convection-diffusion two-point boundary-value problems
\[ -\varepsilon u'' - b u' = f, \quad (0,1), \quad u(0) = g_0, \quad u(1) = g_1. \]
It has been shown that the accuracy of the standard FEM depends crucially on the uniformity of the grid away from the boundary layer. Here, the authors develop a new SDFEM based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property. Optimal convergence results for the standard FEM and the new SDFEM are obtained.

MSC:
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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[1] Bakhalov N.S. (1969). Towards optimization of methods for solving boundary value problems in the presence of boundary layers (in Russian). Zh. Vychisl. Mater. Mater. Fiz. 9: 841–859
[2] Borouchaki, H., Castro-Diaz, M.J., George, P.L., Hecht, F., Mohammadi, B.: Anisotropic adaptive mesh generation in two dimensions for CFD. In: 5th International Conference On Numerical Grid Generation in Computational Field Simulations, vol.3, pp.197–206. Mississppi State University (1996)
[3] Brezzi F., Hughes T.J.R., Marini L.D., Russo A. and Süli E. (1999). A priori error analysis of residual-free bubbles for advection-diffusion problems. SIAM J. Numer. Anal. 36(4): 1933–1948 · Zbl 0947.65115
[4] Brezzi F., Marini D. and Süli E. (2000). Residual-free bubbles for advection-diffusion problems: the general error analysis. Numer. Math. 85: 31–47 · Zbl 0963.65109
[5] Brezzi F. and Russo A. (1994). Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4: 571–587 · Zbl 0819.65128
[6] Cao W., Huang W. and Russell R.D. (1999). A study of monitor functions for two dimensional adaptive mesh generation. SIAM J. Sci. Comput. 20: 1978–1994 · Zbl 0937.65104
[7] Carey G.F. and Dinh H.T. (1985). Grading functions and mesh redistribution. SIAM J. Numer. Anal. 22(5): 1028–1040 · Zbl 0577.65076
[8] Chen, L.: Mesh smoothing schemes based on optimal Delaunay triangulations. In: 13th International Meshing Roundtable, pp.109–120. Sandia National Laboratories, Williamsburg (2004)
[9] Chen L. (2005). New analysis of the sphere covering problems and optimal polytope approximation of convex bodies. J. Approx. Theory 133(1): 134–145 · Zbl 1072.65021
[10] Chen, L., Sun, P., Xu, J.: Multilevel homotopic adaptive finite element methods for convection dominated problems. In: The Proceedings for 15th Conferences for Domain Decomposition Methods. Lecture Notes in Computational Science and Engineering 40, pp.459–468. Springer, Heidelberg (2004) · Zbl 1067.65122
[11] Chen L., Sun P. and Xu J. (2007). Optimal anisotropic simplicial meshes for minimizing interpolation errors in L p -norm. Math. Comput. 76: 179–204 · Zbl 1106.41013
[12] Chen L. and Xu J. (2004). Optimal Delaunay triangulations. J. Comput. Math. 22(2): 299–308 · Zbl 1048.65020
[13] Chen, L., Xu, J.: An optimal streamline diffusion finite element method for a singularly perturbed problem. In: AMS Contemporary Mathematics Series: Recent Advances in Adaptive Computation, vol.383, pp.236–246, Hangzhou (2005) · Zbl 1097.65082
[14] Chen, Y.: Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution. In: Proceedings of the 6th japan-china joint seminar on numerical mathematics (tsukuba, 2002). J. Comput. Appl. Math. 159(1), 25–34 (2003)
[15] Chen Y. (2006). Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid. Adv. Comput. Math. 24: 197–212 · Zbl 1095.65065
[16] D’Azevedo E.F. and Simpson R.B. (1989). On optimal interpolation triangle incidences. SIAM J. Sci. Statist. Comput. 6: 1063–1075 · Zbl 0705.41001
[17] de Boor C. (1973). Good approximation by splines with variable knots. Int. Seines Numer. Math, Birkhauser Verlag, Basel 21: 57–72 · Zbl 0255.41007
[18] Boor C. de(1974). Good approximation by splines with variables knots II. In: Watson, G.A. (eds) Proceedings of the Eleventh International Conference on Numerical Methods in Fluid Dynamics, vol. 363., pp 12–20. Springer, Dundee
[19] Devore R.A. and Lorentz G.G. (1993). Constructive Approximation. Springer, New York · Zbl 0797.41016
[20] Dolejšì V. and Felcman J. (2004). Anisotropic mesh adaptation for numerical solution of boundary value problems. Numer. Methods Partial Differ. Equ. 20: 576–608 · Zbl 1060.65125
[21] Farrell P.A., Hegarty A.F., Miller J.J.H., O’Riordan E. and Shishkin G.I. (2004). Singularly perturbed convection-diffusion problems with boundary and weak interior layers. J. Comput. Appl. Math. 166: 131–151 · Zbl 1041.65059
[22] Franca L.P. and Russo A. (1996). Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles. Appl. Math. Lett. 9: 83–88 · Zbl 0903.65082
[23] Habashi, W.G., Fortin, M., Dompierre, J., Vallet, M.G., Ait-Ali-Yahia, D., Bourgault, Y., Robichaud, M.P., Tam, A., Boivin, S.: Anisotropic mesh optimization for structured and unstructured meshes. In: 28th Computational Fluid Dynamics Lecture Series. von Karman Institute (1997)
[24] Huang W. (2001). Practical aspects of formulation and solution of moving mesh partial differential equations. J. Comput. Phys. 171: 753–775 · Zbl 0990.65107
[25] Huang W. (2001). Variational mesh adaptation: isotropy and equidistribution. J. Comput. Phys. 174: 903–924 · Zbl 0991.65131
[26] Huang W. and Sun W. (2003). Variational mesh adaptation. II: Error estimates and monitor functions. J. Comput. Phys. 184: 619–648 · Zbl 1018.65140
[27] Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the dirichlet-to-neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng, pp.127, no. 1–4, 387–401 (1995) · Zbl 0866.76044
[28] Hughes T.J.R. and Brooks A. (1979). A multidimensional upwind scheme with no crosswind diffusion. In: Hughes, T.J.R. (eds) Finite Element Methods for Convection Dominated Flows, AMD, vol. 34, pp 19–35. ASME, New York
[29] Hughes T.J.R., Feijoo G., Mazzei L. and Quincy J.B. (1998). The variational multiscale method–a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166: 3–24 · Zbl 1017.65525
[30] Johnson C. and Nvert U. (1981). An analysis of some finite element methods for advection-diffusion problems. In: Axelsson, O., Frank, L.S. and Vander Sluis, A. (eds) Analytical and Numerical Approaches to Asymptotic Problems in Analysis., pp 99–116. Amsterdam, NorthHolland
[31] Johnson C., Schatz A.H. and Wahlbin L.B. (1987). Crosswind smear and pointwise errors in streamline diffusion finite element methods. Math. Comput. 49: 25–38 · Zbl 0629.65111
[32] Kellogg R.B. and Tsan A. (1978). Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput. 32: 1025–1039 · Zbl 0418.65040
[33] Kopteva N.V. (1999). Uniform convergence with respect to a small parameter of a scheme with central difference on refining grids. Comput. Math. Math. Phys. 39(10): 1594–1610 · Zbl 0977.65064
[34] Kopteva N.V. (2001). Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem. SIAM J. Numer. Anal. 39(2): 423–441 · Zbl 1003.65091
[35] Kopteva N.V. and Stynes M. (2001). A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem. SIAM J. Numer. Anal. 39: 1446–1467 · Zbl 1012.65076
[36] Lenferink W. (2000). Pointwise convergence of approximations to a convection-diffusion equation on a Shishkin mesh. Appl. Numer. Math. 32(1): 69–86 · Zbl 0942.65090
[37] Linß T. (2001). Sufficient conditions for uniform convergence on layer-adapted grids. Appl. Numer. Math. 37: 241–255 · Zbl 0976.65084
[38] Linß T. (2003). Layer-adapted meshes for convectioni-diffusion problems. Comput. Methods Appl. Mech. Eng. 192: 1061–1105 · Zbl 1022.76036
[39] Linß T. and Stynes M. (2001). The SDFEM on Shishkin meshes for linear convection-diffusion problems. Numer. Math. 87: 457–484 · Zbl 0969.65106
[40] Miller J.J.H., O’Riordan E. and Shishkin G.I. (1995). On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems. IMA J. Numer. Anal. 15(1): 89–99 · Zbl 0814.65082
[41] Miller J.J.H., O’Riordan E. and Shishkin G.I. (1996). Fitted Numerical Methods For Singular Perturbation Problems. World Scientific, Singapore
[42] Morton K.W. (1996). Numerical Solution of Convection-Diffusion Problems, volume 12 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London · Zbl 0861.65070
[43] Nadler E. (1986). Piecewise linear best L 2 approximation on triangulations. In: Chui, C.K., Schumaker, L.L. and Ward, J.D. (eds) Approximation Theory, vol. V, pp 499–502. Academic, New York · Zbl 0616.41023
[44] Niijima K. (1990). Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math. 56: 707–719 · Zbl 0691.65077
[45] Qiu Y., Sloan D.M. and Tang T. (2000). Convergence analysis of an adaptive finite difference method for a singular perturbation problem. J. Comput. Appl. Math. 116: 121–143 · Zbl 0977.65069
[46] Roos H.G. (1998). Layer-adapted grids for singular perturbation problems. ZAMM, Z. Angew. Math. Mech. 78(5): 291–309 · Zbl 0905.65095
[47] Roos H.G., Stynes M. and Tobiska L. (1996). Numerical Methods for Singularly Perturbed Differential Equations, volume 24 of Springer series in Computational Mathematics. Springer, Heidelberg · Zbl 0844.65075
[48] Roos H.G. and Zarin H. (2003). The streamline-diffusion method for a convection-diffusion problem with a point source. J. Comput. Appl. Math. 150: 109–128 · Zbl 1016.65054
[49] Sangalli G. (2003). Quasi optimality of the supg method for the one-dimensional adavection-diffusion problem. SIAM J. Numer. Anal. 41(4): 1528–1542 · Zbl 1058.65080
[50] Schatz A.H. and Wahlbin L.B. (1982). On the quasi-optimality in L of the \(\overset{\circ}{H^1}\) -projection into finite element spacesMath. Comput. 38(157): 1–22 · Zbl 0483.65006
[51] Schatz A.H. and Wahlbin L.B. (1983). On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions. Math. Comput. 40(161): 47–89 · Zbl 0518.65080
[52] Shishkin, G.I.: Grid approximation of singularly perturbed elliptic and parabolic equations (in Russian). PhD thesis, Second doctorial thesis, Keldysh Institute, Moscow (1990) · Zbl 0816.65070
[53] Stynes M. and Tobiska L. (1998). A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numer. Algorithms 18: 337–360 · Zbl 0916.65108
[54] Stynes M. and Tobiska L. (2003). The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis, enhancement of accuracy. SIAM J. Numer. Anal. 41(5): 1620–1642 · Zbl 1055.65121
[55] White A.B. (1979). On selection of equidistributing meshes for two-point boundary-value problems. SIAM J. Numer. Anal. 16: 472–502 · Zbl 0407.65036
[56] Zhang Z.M. (2002). Finite element superconvergence approximation for one-dimensional singularly perturbed problems. Numer. Meth. PDEs 18: 374–395 · Zbl 1002.65088
[57] Zhang Z.M. (2003). Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems. Math. Comput. 72(243): 1147–1177 · Zbl 1019.65091
[58] Zhou G. and Rannacher R. (1996). Pointwise superconvergence of the streamline diffusion finite element method. Numer. Meth. PDEs 12, CMP 96(05): 123–145 · Zbl 0841.65092
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