The finite volume element method on the Shishkin mesh for a singularly perturbed reaction-diffusion problem

Abstract In this paper, we construct a finite volume element method (FVEM) on the Shishkin mesh for solving a singularly perturbed reaction–diffusion problem. The stability of the method is established in energy norm. Further, we derive the optimal error estimate in energy norm, under the decomposition of solution. Some numerical experiments are provided to confirm our theoretical results.

[1]  Wang,et al.  MODIFIED MORLEY ELEMENT METHOD FOR A FOURTH ORDER ELLIPTIC SINGULAR PERTURBATION PROBLEM , 2006 .

[2]  G. Mühlbach.  ECT-B-splines defined by generalized divided differences , 2006 .

[3]  Runchang Lin,et al.  A Balanced Finite Element Method for Singularly Perturbed Reaction-Diffusion Problems , 2012, SIAM J. Numer. Anal..

[4]  Chao Luo,et al.  Generalized Julia sets from a non-analytic complex mapping , 2006, Appl. Math. Comput..

[5]  Junliang Lv,et al.  L2 error estimate of the finite volume element methods on quadrilateral meshes , 2010, Adv. Comput. Math..

[6]  C. Bacuta,et al.  Saddle point least squares for the reaction–diffusion problem , 2020 .

[7]  M. Karimi,et al.  To Investigation the Relationship between Monetary, Fiscal Policy and Economic Growth in Iran: Autoregressive Distributed Lag Approach to Cointegration , 2010 .

[8]  Niall Madden,et al.  A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction–diffusion problems , 2003 .

[9]  Manoj Kumar,et al.  Methods for solving singular perturbation problems arising in science and engineering , 2011, Math. Comput. Model..

[10]  Fang Liu,et al.  A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions , 2009 .

[11]  Shuo Zhang,et al.  Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions , 2015, 1507.04602.

[12]  Martin Stynes,et al.  An analysis of a cell-vertex finite volume method for a parabolic convection-diffusion problem , 1997, Math. Comput..

[13]  Zhimin Zhang,et al.  A Family of Finite Volume Schemes of Arbitrary Order on Rectangular Meshes , 2014, J. Sci. Comput..

[14]  Yh,et al.  GENERALIZED DIFFERENCE METHODS ON ARBITRARY QUADRILATERAL NETWORKS , 1999 .

[15]  Natalia Kopteva,et al.  Shishkin meshes in the numerical solution of singularly perturbed differential equations , 2010 .

[16]  I. M. Navon,et al.  Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems I: Reaction-diffusion Type , 1998, Computers & Mathematics with Applications.

[17]  Martin Stynes,et al.  Convergence analysis of the Adini element on a Shishkin mesh for a singularly perturbed fourth-order problem in two dimensions , 2019, Adv. Comput. Math..

[18]  Zhimin Zhang,et al.  Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems , 2003, Math. Comput..

[19]  Lutz Tobiska,et al.  The SDFEM for a Convection-Diffusion Problem with a Boundary Layer: Optimal Error Analysis and Enhancement of Accuracy , 2003, SIAM J. Numer. Anal..

[20]  E. Bertolazzi,et al.  A CELL-CENTERED SECOND-ORDER ACCURATE FINITE VOLUME METHOD FOR CONVECTION–DIFFUSION PROBLEMS ON UNSTRUCTURED MESHES , 2004 .

[21]  L. Segel,et al.  Introduction to Singular Perturbations. By R. E. O'MALLEY, JR. Academic Press, 1974. $ 16.50. , 1975, Journal of Fluid Mechanics.

[22]  Shaochun Chen,et al.  Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems , 2008, J. Comput. Appl. Math..

[23]  Martin Stynes,et al.  A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion , 1986 .

[24]  Hailong Guo,et al.  Superconvergence of conforming finite element for fourth‐order singularly perturbed problems of reaction diffusion type in 1D , 2014 .

[25]  Zhimin Zhang,et al.  Finite element superconvergence approximation for one‐dimensional singularly perturbed problems , 2002 .

[26]  Martin Stynes,et al.  A Uniformly Convergent Galerkin Method on a Shishkin Mesh for a Convection-Diffusion Problem☆ , 1997 .

[27]  Carmelo Clavero,et al.  A parameter robust numerical method for a two dimensional reaction-diffusion problem , 2005, Math. Comput..

[28]  G. I. Shishkin,et al.  A difference scheme for a singularly perturbed equation of parabolic type with discontinuous boundary conditions , 1990 .

[29]  Endre Süli.  Convergence of finite volume schemes for Poisson's equation on nonuniform meshes , 1991 .

[30]  Panayot S. Vassilevski,et al.  Finite volume methods for convection-diffusion problems , 1996 .

[31]  Jichun Li.  Convergence and superconvergence analysis of finite element methods on highly nonuniform anisotropic meshes for singularly perturbed reaction—diffusion problems , 2001 .

[32]  Tao Lin,et al.  On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials , 2001, SIAM J. Numer. Anal..