Finite volume approximation of stiff problems on two-dimensional curvilinear domain

References

• I. Babuška and J.M. Melenk, The partition of unity method, Int. J. Numer. Meth. Eng. 40 (1997), pp. 727–758. doi: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N
   Web of Science ®Google Scholar
• W. Cheng and R. Temam, Numerical approximation of one-dimensional stationary diffusion equations with boundary layers, Comput. Fluids 31 (2002), pp. 453–466. doi: 10.1016/S0045-7930(01)00060-3
   Web of Science ®Google Scholar
• W. Cheng, R. Temam and X. Wang, New approximation algorithms for a class of partial differential equations displaying boundary layer behavior, Dedicated to Cathleen Morawetz on her 75th birthday, Meth. Appl. Anal. 7 (2000), pp. 363–390.
   Google Scholar
• B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), pp. 173–261. doi: 10.1023/A:1012873910884
   Google Scholar
• B. Cockburn, F. Coquel and P.G. LeFloch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), pp. 687–705. doi: 10.1137/0732032
   Web of Science ®Google Scholar
• B. Cockburn, G. Kanschat and D. Schotzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comput. 74 (2005), pp. 1067–1095. doi: 10.1090/S0025-5718-04-01718-1
   Web of Science ®Google Scholar
• W. Eckhaus, Boundary layers in linear elliptic singular perturbation problems, SIAM Rev. 14 (1972), pp. 225–270. doi: 10.1137/1014030
   Web of Science ®Google Scholar
• W. Eckhaus and E.M. De Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rational Mech. Anal. 23 (1966), pp. 26–86. doi: 10.1007/BF00281135
   Web of Science ®Google Scholar
• R. Eymard, T. Gallouet and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, vol. VII, North-Holland, 2000, pp. 713–1020.
   Google Scholar
• S. Faure, D. Pham and R. Temam, Comparison of finite volume and finite difference methods and application, Anal. Appl. 4 (2006), pp. 163–208. doi: 10.1142/S0219530506000723
   Web of Science ®Google Scholar
• G. Gie, Singular perturbation problems in a general smooth domain, Asymptot. Anal. 6 (2009), pp. 227–249.
   Google Scholar
• G. Gie, M. Hamouda and R. Temam, Boundary layers in smooth curvilinear domains: parabolic problems, Discrete Contin. Dyn. Syst. 26 (2010), pp. 1213–1240.
   Web of Science ®Google Scholar
• M. Hamouda, C. Jung and R. Temam, Boundary layers for the 2D linearized primitive equations, Comm. Pure Appl. Anal. 8 (2009), pp. 335–359.
   Web of Science ®Google Scholar
• H. Han and R.B. Kellogg, A method of enriched subspaces for the numerical solution of a parabolic singular perturbation problem, Comput. Asymp. Meth. Boun. Inter. Layer 4 (1982), pp. 46–52. Dublin.
   Google Scholar
• C.-Y. Jung, Numerical approximation of two-dimensional convection-diffusion equations with boundary layers, Numer. Meth. Part. Diff. Eq. 21 (2005), pp. 623–648. doi: 10.1002/num.20054
   Web of Science ®Google Scholar
• C. Jung and R. Temam, Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers, Internat. J. Numer. Anal. Model. 2 (2005), pp. 367–408.
   Web of Science ®Google Scholar
• C. Jung and R. Temam, Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point, J. Math. Phys. 48 (2007), p. 27. doi: 10.1063/1.2347899
   Web of Science ®Google Scholar
• C.Y. Jung and R. Temam, Finite volume approximation of one-dimensional stiff convection-diffusion equations, J. Sci. Comput. 41 (2009), pp. 384–410. doi: 10.1007/s10915-009-9304-2
   Web of Science ®Google Scholar
• C. Jung and R. Temam, Finite volume approximation of two-dimensional stiff problems, INJAM 7 (2010), pp. 462–476.
   Web of Science ®Google Scholar
• R.B. Kellogg and M. Stynes, Layers and corner singularities in singularly perturbed elliptic problems, BIT 48 (2008), pp. 309–314. doi: 10.1007/s10543-008-0173-0
   Web of Science ®Google Scholar
• A.L. Mazzucato, V. Nistor and Q. Qu, A nonconforming generalized finite element method for transmission problems, SIAM J. Numer. Anal. 51 (2013), pp. 555–576. doi: 10.1137/100816031
   Web of Science ®Google Scholar
• A.L. Mazzucato, V. Nistor and Q. Qu, Quasi-optimal rates of convergence for the generalized finite element method in polygonal domains, J. Comput. Appl. Math. 263 (2014), pp. 466–477. doi: 10.1016/j.cam.2013.12.026
   Web of Science ®Google Scholar
• N. Moës, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing, Int. J. Numer. Meth. Eng. 46 (1999), pp. 131–151. doi: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
   Web of Science ®Google Scholar
• R.E. O'Malley, Singular perturbation analysis for ordinary differential equations, Commun. Math. Inst., Rijksuniversiteit Utrecht 5 (1977).
   Google Scholar
• H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 1996.
   Google Scholar
• S.-D. Shih and R.B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal. 18 (1987), pp. 1467–1511. doi: 10.1137/0518107
   Web of Science ®Google Scholar
• M. Stynes, Steady-state convection-diffusion problems, Acta. Numer. 14 (2005), pp. 445–508. doi: 10.1017/S0962492904000261
   Google Scholar