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Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems. (English) Zbl 1203.65261
Summary: We study the stability and accuracy of adaptive finite element methods for the convection-dominated convection-diffusion-reaction problem in the two-dimension space. Through various numerical examples on a type of layer-adapted grids (Shishkin grids), we show that the mesh adaptivity driven by accuracy alone cannot stabilize the scheme in all cases. Furthermore the numerical approximation is sensitive to the symmetry of the grid in the region where the solution is smooth. On the basis of these two observations, we develop a multilevel-homotopic-adaptive finite element method (MHAFEM) by combining streamline diffusion finite element method, anisotropic mesh adaptation, and the homotopy of the diffusion coefficient. We use numerical experiments to demonstrate that MHAFEM can efficiently capture boundary or interior layers and produce accurate solutions.
MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
References:
[1]Bank, R., Bürger, J., Fichtner, W., Smith, R.: Some upwinding techniques for finite element approximations of convection diffusion equations. Numer. Math. 58, 185–202 (1990) · Zbl 0713.65066 · doi:10.1007/BF01385618
[2]Bank, R.E., Smith, R.K.: Mesh smoothing using a posteriori error estimates. SIAM J. Numer. Anal. 34, 979–997 (1997) · Zbl 0873.65092 · doi:10.1137/S0036142994265292
[3]Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part I: Grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294–2312 (2003) · Zbl 1058.65116 · doi:10.1137/S003614290139874X
[4]Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part II: General unstructured grids. SIAM J. Numer. Anal. 41(6), 2313–2332 (2003) · Zbl 1058.65117 · doi:10.1137/S0036142901398751
[5]Bänsch, E., Morin, P., Nochetto, R.H.: An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition. SIAM J. Numer. Anal. 40(4), 1207–1229 (2002) · Zbl 1027.65148 · doi:10.1137/S0036142901392134
[6]Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 175, 311–341 (1999) · Zbl 0924.76051 · doi:10.1016/S0045-7825(98)00359-4
[7]Brezzi, F., Franca, L., Russo, A.: Further considerations on residual free bubbles for advection-diffusive equations. Comput. Methods Appl. Mech. Eng. 166, 25–33 (1998) · Zbl 0934.65126 · doi:10.1016/S0045-7825(98)00080-2
[8]Brezzi, F., Franca, L.P., Hughes, T.J.R., Russo, A.: b=g. Comput. Methods Appl. Mech. Eng. 145, 329–339 (1997) · Zbl 0904.76041 · doi:10.1016/S0045-7825(96)01221-2
[9]Brezzi, F., Hughes, T.J.R., Marini, L.D., Russo, A., Süli, E.: A priori error analysis of residual-free bubbles for advection-diffusion problems. SIAM J. Numer. Anal. 36(4), 1933–1948 (1999) · Zbl 0947.65115 · doi:10.1137/S0036142998342367
[10]Brezzi, F., Marini, D., Süli, E.: Residual-free bubbles for advection-diffusion problems: the general error analysis. Numer. Math. 85, 31–47 (2000) · Zbl 0963.65109 · doi:10.1007/s002110050476
[11]Brezzi, F., Marini, L.D., Pietra, P.: Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26(6), 1342–1355 (1989) · Zbl 0686.65088 · doi:10.1137/0726078
[12]Brooks, A., Hughes, T.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 32, 199–259 (1982) · Zbl 0497.76041 · doi:10.1016/0045-7825(82)90071-8
[13]Chen, L.: Mesh smoothing schemes based on optimal Delaunay triangulations. In: 13th International Meshing Roundtable, Williamsburg, VA, 2004, pp. 109–120. Sandia National Laboratories, Albuquerque (2004)
[14]Chen, L.: Robust and accurate algorithms for solving anisotropic singularities. PhD thesis, Department of Mathematics, The Pennsylvania State University (2005)
[15]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, vol. 40, pp. 459–468. Springer, Berlin (2004)
[16]Chen, L., Sun, P., Xu, J.: Optimal anisotropic simplicial meshes for minimizing interpolation errors in L p -norm. Math. Comput. 76(257), 179–204 (2007) · Zbl 1106.41013 · doi:10.1090/S0025-5718-06-01896-5
[17]Chen, L., Xu, J.: Stability and accuracy of adapted finite element methods for singularly perturbed problems. Numer. Math. 109(2), 167–191 (2008) · Zbl 1146.65059 · doi:10.1007/s00211-007-0118-6
[18]Heinrich, J.C., Huyakorn, P.S., Zienkiewicz, O.C., Mitchell, A.R.: An ’upwind’ finite element scheme for two-dimensional convective transport equation. Int. J. Numer. Methods Eng. 11, 131–143 (1977) · Zbl 0353.65065 · doi:10.1002/nme.1620110113
[19]Hemker, P.W.: A singularly perturbed model problem for numerical computation. J. Comput. Appl. Math. 76(1–2), 277–285 (1996) · Zbl 0870.35020 · doi:10.1016/S0377-0427(96)00113-6
[20]Holmes, M.H.: Introduction to Perturbation Methods. Texts in Applied Mathematics, vol. 20. Springer, New York (1995)
[21]Houston, P., Schwab, C., Suli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002) · Zbl 1015.65067 · doi:10.1137/S0036142900374111
[22]Huang, W.: Mathematical principles of anisotropic mesh adaptation. Commun. Comput. Phys. 1, 276–310 (2006)
[23]Huang, W., Sun, W.: Variational mesh adaptation II: error estimates and monitor functions. J. Comput. Phys. 184, 619–648 (2003) · Zbl 1018.65140 · doi:10.1016/S0021-9991(02)00040-2
[24]Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Hughes, T.J.R. (ed.) Finite Element Methods for Convection Dominated Flows. AMD, vol. 34, pp. 19–35. ASME, New York (1979)
[25]John, V.: A numerical study of a posteriori error estimators for convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 190(5–7), 757–781 (2000) · Zbl 0973.76049 · doi:10.1016/S0045-7825(99)00440-5
[26]John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: part I–a review. Comput. Methods Appl. Mech. Eng. 196(17–20), 2197–2215 (2007) · Zbl 1173.76342 · doi:10.1016/j.cma.2006.11.013
[27]John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: part II–analysis for P1 and Q1 finite elements. Comput. Methods Appl. Mech. Eng. 197, 1997–2014 (2008) · Zbl 1194.76122 · doi:10.1016/j.cma.2007.12.019
[28]Kang, T., Yu, D.: Some a posteriori error estimates of the finite-difference streamline-diffusion method for convection-dominated diffusion equations. Adv. Comput. Math. 15, 193–218 (2001) · Zbl 0990.65103 · doi:10.1023/A:1014294002686
[29]Li, R., Tang, T., Zhang, P.: Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys. 170(2), 562–588 (2001) · Zbl 0986.65090 · doi:10.1006/jcph.2001.6749
[30]Li, R., Tang, T., Zhang, P.: A moving mesh finite element algorithm for singular problems in two and three space dimensions. J. Comput. Phys. 177(2), 365–393 (2002) · Zbl 0998.65105 · doi:10.1006/jcph.2002.7002
[31]Linß, T.: Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem. IMA J. Numer. Anal. 20, 621–632 (2000) · Zbl 0966.65083 · doi:10.1093/imanum/20.4.621
[32]Linß, T.: The necessity of Shishkin-decompositions. Appl. Math. Lett. 14, 891–896 (2001) · Zbl 0986.65071 · doi:10.1016/S0893-9659(01)00061-1
[33]Linß, T.: Layer-adapted meshes for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 192, 1061–1105 (2003) · Zbl 1022.76036 · doi:10.1016/S0045-7825(02)00630-8
[34]Linß, T., Stynes, M.: Numerical methods on Shishkin meshes for linear convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 190(28), 3527–3542 (2001) · Zbl 0988.76062 · doi:10.1016/S0045-7825(00)00271-1
[35]Linß, T., Stynes, M.: The SDFEM on Shishkin meshes for linear convection-diffusion problems. Numer. Math. 87, 457–484 (2001) · doi:10.1007/PL00005420
[36]Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems. IMA J. Numer. Anal. 15, 89–99 (1995) · Zbl 0814.65082 · doi:10.1093/imanum/15.1.89
[37]Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore (1996)
[38]Morton, K.W.: Numerical Solution of Convection-Diffusion Problems. Applied Mathematics and Mathematical Computation, vol. 12. Chapman & Hall, London (1996)
[39]Nooyen, R.R.P.V.: A Petrov-Galerkin mixed finite element method with exponential fitting. Numer. Methods Partial Differ. Equ. 11(5), 501–524 (1995) · Zbl 0840.65130 · doi:10.1002/num.1690110507
[40]Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, vol. 24. Springer, Berlin (1996)
[41]Roos, H.G.R.G.: A note on the conditioning of upwind schemes on Shishkin meshes. IMA J. Numer. Anal. 16, 529 (1996) · Zbl 0861.65066 · doi:10.1093/imanum/16.4.529
[42]Shishkin, G.I.: Grid approximation of singularly perturbed elliptic and parabolic equations. PhD thesis, Second doctoral thesis, Keldysh Institute, Moscow (1990) (in Russian)
[43]Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005) · Zbl 1115.65108 · doi:10.1017/S0962492904000261
[44]Sun, P., Russell, R.D., Xu, J.: A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem. J. Comput. Phys. 224(2), 1021–1048 (2007) · Zbl 1123.76343 · doi:10.1016/j.jcp.2006.11.005
[45]Tobiska, L.: Analysis of a new stabilized higher order finite element method for advection-diffusion equations. Comput. Methods Appl. Mech. Eng. 196, 538–550 (2006) · Zbl 1120.76336 · doi:10.1016/j.cma.2006.05.009
[46]Xu, J., Zikatanov, L.: A monotone finite element scheme for convection diffusion equations. Math. Comput. 68, 1429–1446 (1999) · Zbl 0931.65111 · doi:10.1090/S0025-5718-99-01148-5
[47]Zhang, Z., Tang, T.: An adaptive mesh redistribution algorithm for convection-dominated problems. Commun. Pure Appl. Anal. 1(3), 341–357 (2002) · Zbl 1008.65091 · doi:10.3934/cpaa.2002.1.341
[48]Zhang, Z.M.: 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