Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities. (English) Zbl 1163.65087
The author develops a discontinuous Galerkin least-squares finite element method for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities by recasting the second-order elliptic equations as a system of first-order equations. Coercivity and uniform error estimates for the finite element approximation are established in an appropriately scald form. Numerical examples that confirm the theoretical results are also provided.
Reviewer: Răzvan Răducanu (Iaşi)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B25 | Singular perturbations in context of PDEs |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
reaction-diffusion problems; discontinuous coefficients; boundary singularities; singular perturbation; discontinuous Galerkin least-squares finite element method; second-order elliptic equations; error estimates; numerical examplesReferences:
| [1] | Adams R.A.: Sobolev Spaces. Academic Press, New York (1975) · Zbl 0314.46030 |
| [2] | Arnold D.N., Brezzi F., Cockburn B., Marini L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002) · Zbl 1008.65080 · doi:10.1137/S0036142901384162 |
| [3] | Babuška I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970) · Zbl 0199.50603 · doi:10.1007/BF02248021 |
| [4] | Babuška I., Baumann C.E., Oden J.T.: A discontinuous hp finite element method for diffusion problems: 1-D analysis. Comput. Math. Appl. 37, 103–122 (1999) · Zbl 0940.65076 · doi:10.1016/S0898-1221(99)00117-0 |
| [5] | Bensow R.E., Larson M.G.: Discontinuous/continuous least-squares finite element methods for elliptic problems. Math. Models Methods Appl. Sci. 15, 825–842 (2005) · Zbl 1085.65107 · doi:10.1142/S0218202505000595 |
| [6] | Bensow R.E., Larson M.G.: Discontinuous least-squares finite element method for the div-curl problem. Numer. Math. 101, 601–617 (2005) · Zbl 1083.65104 · doi:10.1007/s00211-005-0600-y |
| [7] | Bernardi C., Verfürth R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579–608 (2000) · Zbl 0962.65096 · doi:10.1007/PL00005393 |
| [8] | Berndt M., Manteuffel T.A., McCormick S.F., Starke G.: Analysis of first-order system least squares (FOSLS) for elliptic problems with discontinuous coefficients. I. SIAM J. Numer. Anal. 43, 386–408 (2005) · Zbl 1087.65115 · doi:10.1137/S0036142903427688 |
| [9] | Berndt M., Manteuffel T.A., McCormick S.F.: Analysis of first-order system least squares (FOSLS) for elliptic problems with discontinuous coefficients. II. SIAM J. Numer. Anal. 43, 409–436 (2005) · Zbl 1087.65114 · doi:10.1137/S003614290342769X |
| [10] | Bjørstad P.E., Dryja M., Rahman T.: Additive Schwarz methods for elliptic mortar finite element problems. Numer. Math. 95, 427–457 (2003) · Zbl 1036.65107 · doi:10.1007/s00211-002-0429-6 |
| [11] | Bochev P.B., Gunzburger M.D.: Finite element methods of least-squares type. SIAM Rev. 40, 789–837 (1998) · Zbl 0914.65108 · doi:10.1137/S0036144597321156 |
| [12] | Bramble J.H., Lazarov R.D., Pasciak J.E.: A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comput. 66, 935–955 (1997) · Zbl 0870.65104 · doi:10.1090/S0025-5718-97-00848-X |
| [13] | Bramble J.H., Lazarov R.D., Pasciak J.E.: Least-squares methods for linear elasticity based on a discrete minus one inner product. Comput. Methods Appl. Mech. Eng. 191, 727–744 (2001) · Zbl 0999.74107 · doi:10.1016/S0045-7825(01)00255-9 |
| [14] | Brayanov I.A.: Numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem with discontinuous coefficients. Appl. Math. Comput. 182, 631–643 (2006) · Zbl 1113.65106 · doi:10.1016/j.amc.2006.04.027 |
| [15] | Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994) · Zbl 0804.65101 |
| [16] | Cai Z., Lazarov R.D., Manteuffel T.A., McCormick S.F.: First-order system least squares for second-order partial differential equations. I. SIAM J. Numer. Anal. 31, 1785–1799 (1994) · Zbl 0813.65119 · doi:10.1137/0731091 |
| [17] | Cai Z., Manteuffel T.A., McCormick S.F.: First-order system least squares for second-order partial differential equations. II. SIAM J. Numer. Anal. 34, 425–454 (1997) · Zbl 0912.65089 · doi:10.1137/S0036142994266066 |
| [18] | Cai Z., Manteuffel T.A., McCormick S.F., Ruge J.: First-order system \({\mathcal{LL}^*}\) (FOSLL*): scalar elliptic partial differential equations. SIAM J. Numer. Anal. 39, 1418–1445 (2001) · Zbl 1008.65085 · doi:10.1137/S0036142900388049 |
| [19] | Cai Z., Westphal C.R.: A weighted H(div) least-squares method for second-order elliptic problems. SIAM J. Numer. Anal. 46, 1640–1651 (2008) · Zbl 1168.65069 · doi:10.1137/070698531 |
| [20] | Cao Y., Gunzburger M.D.: Least-squares finite element approximations to solutions of interface problems. SIAM J. Numer. Anal. 35, 393–405 (1998) · Zbl 0913.65096 · doi:10.1137/S0036142996303249 |
| [21] | Chen Z., Dai S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24, 443–462 (2002) · Zbl 1032.65119 · doi:10.1137/S1064827501383713 |
| [22] | Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) · Zbl 0383.65058 |
| [23] | Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.): Discontinuous galerkin methods. Theory, computation and applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, New York (2000) · Zbl 0935.00043 |
| [24] | Cockburn B., Shu C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998) · Zbl 0927.65118 · doi:10.1137/S0036142997316712 |
| [25] | Cockburn B., Shu C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001) · Zbl 1065.76135 · doi:10.1023/A:1012873910884 |
| [26] | Dokeva N., Dryja M., Proskurowski W.: A FETI-DP preconditioner with a special scaling for mortar discretization of elliptic problems with discontinuous coefficients. SIAM J. Numer. Anal. 44, 283–299 (2006) · Zbl 1113.65108 · doi:10.1137/040616401 |
| [27] | Farrell P.A., Miller J.J.H., O’Riordan E., Shishkin G.I.: Singularly perturbed differential equations with discontinuous source terms. In: Vulkov, L.G., Miller, J.J.H., Shishkin, G.I.(eds) Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, pp. 23–31. Nova Science Publishers, Inc., New York (2000) |
| [28] | Gerritsma M.I., Proot M.M.J.: Analysis of a discontinuous least squares spectal element method. J. Sci. Comput. 17, 297–306 (2002) · Zbl 1004.65073 · doi:10.1023/A:1015173203136 |
| [29] | Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224. Springer, Berlin (1983) · Zbl 0562.35001 |
| [30] | Houston P., Jensen M., Süli E.: hp-discontinuous Galerkin finite element methods with least-squares stabilization. J. Sci. Comput. 17, 3–25 (2002) · Zbl 1053.76040 · doi:10.1023/A:1015180009979 |
| [31] | Huang J., Zou J.: A mortar element method for elliptic problems with discontinuous coefficients. IMA J. Numer. Anal. 22, 549–576 (2002) · Zbl 1014.65117 · doi:10.1093/imanum/22.4.549 |
| [32] | Jiang B.N.: The Least-squares finite element method. Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, Berlin (1998) · Zbl 0904.76003 |
| [33] | Lee E., Manteuffel T.A., Westphal C.R.: Weighted-norm first-order system least squares (FOSLS) for problems with corner singularities. SIAM J. Numer. Anal. 44, 1974–1996 (2006) · Zbl 1129.65087 · doi:10.1137/050636279 |
| [34] | Lee E., Manteuffel T.A., Westphal C.R.: Weighted-norm first-order system least-squares (FOSLS) for div/curl systems with three dimensional edge singularities. SIAM J. Numer. Anal. 46, 1619–1639 (2008) · Zbl 1170.65095 · doi:10.1137/06067345X |
| [35] | Lin R.: Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions. SIAM J. Numer. Anal. 47, 89–108 (2008) · Zbl 1189.65282 · doi:10.1137/070700267 |
| [36] | Manteuffel T.A., McCormick S.F., Ruge J., Schmidt J.G.: First-order system \({\mathcal{LL}^*}\) (FOSLL*) for general scalar elliptic problems in the plane. SIAM J. Numer. Anal. 43, 2098–2120 (2005) · Zbl 1103.65117 · doi:10.1137/S0036142903430402 |
| [37] | Marcinkowski L.: Additive Schwarz method for mortar discretization of elliptic problems with P 1 nonconforming finite elements. BIT 45, 375–394 (2005) · Zbl 1080.65118 · doi:10.1007/s10543-005-7123-x |
| [38] | Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted numerical methods for singular perturbation problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific Publishing Co., Inc., River Edge (1996) · Zbl 0915.65097 |
| [39] | Miller J.J.H., O’Riordan E., Shishkin G.I., Wang S.: A parameter-uniform Schwarz method for a singularly perturbaed reaction-diffusion problem with an interior-layer. Appl. Numer. Math. 35, 323–337 (2000) · Zbl 0967.65086 · doi:10.1016/S0168-9274(99)00140-3 |
| [40] | Morton, K.W.: Numerical solution of convection-diffusion problems. Applied Mathematics and Mathematical Computation, vol. 12. Chapman & Hall, London (1996) · Zbl 0861.65070 |
| [41] | Oden J.T., Babuška I., Baumann C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491–519 (1998) · Zbl 0926.65109 · doi:10.1006/jcph.1998.6032 |
| [42] | Oden, J.T., Carey, G.F.: Finite elements, mathematical aspects. The Texas Finite Element Series, vol IV. Prentice-Hall, New Jersey (1983) · Zbl 0496.65055 |
| [43] | O’Riordan E., Stynes M.: A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions. Math. Comp. 57, 47–62 (1991) · Zbl 0733.65063 |
| [44] | Pontaza J.P., Reddy J.N.: Least-squares finite element formulations for viscous incompressible and compressible fluid flows. Comput. Methods Appl. Mech. Eng. 195, 2454–2494 (2006) · Zbl 1178.76237 · doi:10.1016/j.cma.2005.05.018 |
| [45] | Rahman T., Xu X., Hoppe R.: Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients. Numer. Math. 101, 551–572 (2005) · Zbl 1087.65117 · doi:10.1007/s00211-005-0625-2 |
| [46] | Roos H.-G., Stynes M., Tobiska L.: Numerical methods for singularly perturbed differential equations. Convection-Diffusion and Flow Problems. Springer, Berlin (1996) · Zbl 0844.65075 |
| [47] | Roos H.-G., Zarin H.: A second-order scheme for singularly perturbed differential equations with discontinuous source term. J. Numer. Math. 10, 275–289 (2002) · Zbl 1023.65077 · doi:10.1515/JNMA.2002.275 |
| [48] | Sauter S.A., Warnke R.: Composite finite elements for elliptic boundary value problems with discontinuous coefficients. Computing 77, 29–55 (2006) · Zbl 1284.65173 · doi:10.1007/s00607-005-0150-2 |
| [49] | Stynes M., O’Riordan E.: An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques. Math. Comp. 56, 663–675 (1991) · Zbl 0718.65062 · doi:10.1090/S0025-5718-1991-1068809-4 |
| [50] | Xie Z., Zhang Z.: Superconvergence of DG method for one-dimensional singularly perturbed problems. J. Comput. Math. 25, 185–200 (2007) · Zbl 1142.65388 |
| [51] | Zarin H., Roos H.-G.: Interior penalty discontinuous approximations of convection-diffusion problems with parabolic layers. Numer. Math. 100, 735–759 (2005) · Zbl 1100.65105 · doi:10.1007/s00211-005-0598-1 |
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.
