An adapted Petrov–Galerkin multi-scale finite element method for singularly perturbed reaction–diffusion problems

References

• J.E. Aarnes and Y. Efendiev, Mixed multiscale finite element methods for stochastic porous media flows, SIAM J. Sci. Comput. 30(5) (2008), pp. 2319–2339. doi: 10.1137/07070108X
   Web of Science ®Google Scholar
• I. Babuska and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul. 9(1) (2011), pp. 373–406. doi: 10.1137/100791051
   Web of Science ®Google Scholar
• L. Chen and J.C. Xu, Stability and accuracy of adapted finite element methods for singularly perturbed problems, Numer. Math. 109(2) (2008), pp. 167–191. doi: 10.1007/s00211-007-0118-6
   Web of Science ®Google Scholar
• L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part II. Optimal test functions, Numer. Methods Partial Differential Equations 27(1) (2011), pp. 70–105. doi: 10.1002/num.20640
   Web of Science ®Google Scholar
• Y. Efendiev, J. Galvis, and E. Gildin, Local-global multiscale model reduction for flows in high-contrast heterogeneous media, J. Comput. Phys. 231(24) (2012), pp. 8100–8113. doi: 10.1016/j.jcp.2012.07.032
   Web of Science ®Google Scholar
• Y. Efendiev, J. Galvis, and T.Y. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys. 251 (2013), pp. 116–135. doi: 10.1016/j.jcp.2013.04.045
   Web of Science ®Google Scholar
• Y. Efendiev, J. Galvis, and X.H. Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comput. Phys. 230(4) (2011), pp. 937–955. doi: 10.1016/j.jcp.2010.09.026
   Web of Science ®Google Scholar
• L.P. Franca, A.L. Madureira, and F. Valentin, Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions, Comput. Methods Appl. Mech. Engrg. 194(27–29) (2005), pp. 3006–3021. doi: 10.1016/j.cma.2004.07.029
   Web of Science ®Google Scholar
• S. Franz, T. Linb, H.G. Roos, and S. Schiller, Uniform superconvergence of a finite element method with edge stabilization for convection-diffusion problems, J. Comput. Math. 28(1) (2010), pp. 32–44.
   Web of Science ®Google Scholar
• T.Y. Hou, X.H. Wu, and Z.Q. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comput. 68(227) (1999), pp. 913–943. doi: 10.1090/S0025-5718-99-01077-7
   Web of Science ®Google Scholar
• T.Y. Hou, X.H. Wu, and Y. Zhang, Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation, Commun. Math. Sci. 2(2) (2004), pp. 185–205. doi: 10.4310/CMS.2004.v2.n2.a3
   Google Scholar
• S. Jiang and Y.Q. Huang, Numerical investigation on the boundary conditions for the multiscale base functions, Commun. Comput. Phys. 5(5) (2009), pp. 928–941.
   Web of Science ®Google Scholar
• L.J. Jiang and M. Presho, A resourceful splitting technique with applications to deterministic and stochastic multiscale finite element methods, Multiscale Model. Simul. 10(3) (2012), pp. 954–985. doi: 10.1137/110843253
   Web of Science ®Google Scholar
• N. Kopteva and M. Stynes, A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal. 39(4) (2001), pp. 1446–1467. doi: 10.1137/S003614290138471X
   Web of Science ®Google Scholar
• R. Li, P.B. Ming, and F.Y. Tang, An efficient high order heterogeneous multiscale method for elliptic problems, Multiscale Model. Simul. 10(1) (2012), pp. 259–283. doi: 10.1137/110836626
   Web of Science ®Google Scholar
• R. Li, T. Tang, and P.W. Zhang, A moving mesh finite element algorithm for singular problems in two and three space dimensions, J. Comput. Phys. 177(2) (2002), pp. 365–393. doi: 10.1006/jcph.2002.7002
   Web of Science ®Google Scholar
• R.C. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions, SIAM J. Numer. Anal. 47(1) (2009), pp. 89–108. doi: 10.1137/070700267
   Web of Science ®Google Scholar
• J.J. Miller, E. O'Riordan, and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
   Google Scholar
• P.J. Park and T.Y. Hou, Multiscale numerical methods for singularly perturbed convection-diffusion equations, Int. J. Comput. Method 1(1) (2004), pp. 17–65. doi: 10.1142/S0219876204000071
   Web of Science ®Google Scholar
• H.G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 1996.
   Google Scholar
• F. Su, Z. Xu, J.Z. Cui, X.P. Du, and H. Jiang, Multiscale computation method for an advection-diffusion equation, Appl. Math. Comput. 218(14) (2012), pp. 7369–7374. doi: 10.1016/j.amc.2011.12.001
   Web of Science ®Google Scholar
• P.T. Sun, L. Chen, and J.C. Xu, Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems, J. Sci. Comput. 43(1) (2010), pp. 24–43. doi: 10.1007/s10915-009-9337-6
   Web of Science ®Google Scholar
• Z.Q. Xie, Z.Z. Zhang, and Z.M. Zhang, A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems, J. Comput. Math. 27(2–3) (2009), pp. 280–298.
   Web of Science ®Google Scholar