Abstract
In this paper, a discontinuous Galerkin least-squares finite element method is developed 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. In a companion paper (Lin in SIAM J Numer Anal 47:89–108, 2008) a similar method has been developed for problems with continuous data and shown to be well-posed, uniformly convergent, and optimal in convergence rate. In this paper the method is modified to take care of conditions that arise at interfaces and boundary singularities. Coercivity and uniform error estimates for the finite element approximation are established in an appropriately scaled norm. Numerical examples confirm the theoretical results.
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Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)
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)
Babuška I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970)
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)
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)
Bensow R.E., Larson M.G.: Discontinuous least-squares finite element method for the div-curl problem. Numer. Math. 101, 601–617 (2005)
Bernardi C., Verfürth R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579–608 (2000)
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)
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)
Bjørstad P.E., Dryja M., Rahman T.: Additive Schwarz methods for elliptic mortar finite element problems. Numer. Math. 95, 427–457 (2003)
Bochev P.B., Gunzburger M.D.: Finite element methods of least-squares type. SIAM Rev. 40, 789–837 (1998)
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)
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)
Brayanov I.A.: Numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem with discontinuous coefficients. Appl. Math. Comput. 182, 631–643 (2006)
Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)
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)
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)
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)
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)
Cao Y., Gunzburger M.D.: Least-squares finite element approximations to solutions of interface problems. SIAM J. Numer. Anal. 35, 393–405 (1998)
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)
Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
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)
Cockburn B., Shu C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
Cockburn B., Shu C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)
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)
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)
Gerritsma M.I., Proot M.M.J.: Analysis of a discontinuous least squares spectal element method. J. Sci. Comput. 17, 297–306 (2002)
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)
Houston P., Jensen M., Süli E.: hp-discontinuous Galerkin finite element methods with least-squares stabilization. J. Sci. Comput. 17, 3–25 (2002)
Huang J., Zou J.: A mortar element method for elliptic problems with discontinuous coefficients. IMA J. Numer. Anal. 22, 549–576 (2002)
Jiang B.N.: The Least-squares finite element method. Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, Berlin (1998)
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)
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)
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)
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)
Marcinkowski L.: Additive Schwarz method for mortar discretization of elliptic problems with P 1 nonconforming finite elements. BIT 45, 375–394 (2005)
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)
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)
Morton, K.W.: Numerical solution of convection-diffusion problems. Applied Mathematics and Mathematical Computation, vol. 12. Chapman & Hall, London (1996)
Oden J.T., Babuška I., Baumann C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491–519 (1998)
Oden, J.T., Carey, G.F.: Finite elements, mathematical aspects. The Texas Finite Element Series, vol IV. Prentice-Hall, New Jersey (1983)
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)
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)
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)
Roos H.-G., Stynes M., Tobiska L.: Numerical methods for singularly perturbed differential equations. Convection-Diffusion and Flow Problems. Springer, Berlin (1996)
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)
Sauter S.A., Warnke R.: Composite finite elements for elliptic boundary value problems with discontinuous coefficients. Computing 77, 29–55 (2006)
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)
Xie Z., Zhang Z.: Superconvergence of DG method for one-dimensional singularly perturbed problems. J. Comput. Math. 25, 185–200 (2007)
Zarin H., Roos H.-G.: Interior penalty discontinuous approximations of convection-diffusion problems with parabolic layers. Numer. Math. 100, 735–759 (2005)
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Lin, R. Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities. Numer. Math. 112, 295–318 (2009). https://doi.org/10.1007/s00211-008-0208-0
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DOI: https://doi.org/10.1007/s00211-008-0208-0