October  2015, 8(5): 901-911. doi: 10.3934/dcdss.2015.8.901

Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations

1. 

Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic

Received  February 2014 Revised  August 2014 Published  July 2015

A recently proposed local projection stabilization (LPS) finite element method containing a nonlinear crosswind diffusion term is analyzed for a transient convection-diffusion-reaction equation using a one-step $\theta$-scheme as temporal discretization. Both the fully nonlinear method and its semi-implicit variant are considered. Solvability of the discrete problem is established and a priori error estimates in the LPS norm are proved. Uniqueness of the discrete solution is proved for the semi-implicit approach or for sufficiently small time steps.
Citation: Petr Knobloch. Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 901-911. doi: 10.3934/dcdss.2015.8.901
References:
[1]

G. Barrenechea, V. John and P. Knobloch, A nonlinear local projection stabilization for convection-diffusion-reaction equations, in Numerical Mathematics and Advanced Applications 2011, Proceedings of ENUMATH 2011 (eds. A. Cangiani, R. Davidchack, E. Georgoulis, A. Gorban, J. Levesley and M. Tretyakov), Springer-Verlag, Berlin, 2013, 237-245. doi: 10.1007/978-3-642-33134-3_26.  Google Scholar

[2]

S. Ganesan and L. Tobiska, Stabilization by local projection for convection-diffusion and incompressible flow problems, J. Sci. Comput., 43 (2010), 326-342. doi: 10.1007/s10915-008-9259-8.  Google Scholar

[3]

V. John and P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part I - A review, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2197-2215. doi: 10.1016/j.cma.2006.11.013.  Google Scholar

[4]

P. Knobloch, A generalization of the local projection stabilization for convection-diffusion-reaction equations, SIAM J. Numer. Anal., 48 (2010), 659-680. doi: 10.1137/090767807.  Google Scholar

[5]

P. Knobloch, Local projection method for convection-diffusion-reaction problems with projection spaces defined on overlapping sets, in Numerical Mathematics and Advanced Applications 2009, Proceedings of ENUMATH 2009 (eds. G. Kreiss, P. Lötstedt, A. Målqvist and M. Neytcheva), Springer-Verlag, Berlin, 2010, 497-505. doi: 10.1007/978-3-642-11795-4_53.  Google Scholar

[6]

G. Matthies, P. Skrzypacz and L. Tobiska, A unified convergence analysis for local projection stabilizations applied to the Oseen problem, M2AN Math. Model. Numer. Anal., 41 (2007), 713-742. doi: 10.1051/m2an:2007038.  Google Scholar

[7]

H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems. 2nd ed., Springer-Verlag, Berlin, 2008.  Google Scholar

show all references

References:
[1]

G. Barrenechea, V. John and P. Knobloch, A nonlinear local projection stabilization for convection-diffusion-reaction equations, in Numerical Mathematics and Advanced Applications 2011, Proceedings of ENUMATH 2011 (eds. A. Cangiani, R. Davidchack, E. Georgoulis, A. Gorban, J. Levesley and M. Tretyakov), Springer-Verlag, Berlin, 2013, 237-245. doi: 10.1007/978-3-642-33134-3_26.  Google Scholar

[2]

S. Ganesan and L. Tobiska, Stabilization by local projection for convection-diffusion and incompressible flow problems, J. Sci. Comput., 43 (2010), 326-342. doi: 10.1007/s10915-008-9259-8.  Google Scholar

[3]

V. John and P. Knobloch, On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part I - A review, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2197-2215. doi: 10.1016/j.cma.2006.11.013.  Google Scholar

[4]

P. Knobloch, A generalization of the local projection stabilization for convection-diffusion-reaction equations, SIAM J. Numer. Anal., 48 (2010), 659-680. doi: 10.1137/090767807.  Google Scholar

[5]

P. Knobloch, Local projection method for convection-diffusion-reaction problems with projection spaces defined on overlapping sets, in Numerical Mathematics and Advanced Applications 2009, Proceedings of ENUMATH 2009 (eds. G. Kreiss, P. Lötstedt, A. Målqvist and M. Neytcheva), Springer-Verlag, Berlin, 2010, 497-505. doi: 10.1007/978-3-642-11795-4_53.  Google Scholar

[6]

G. Matthies, P. Skrzypacz and L. Tobiska, A unified convergence analysis for local projection stabilizations applied to the Oseen problem, M2AN Math. Model. Numer. Anal., 41 (2007), 713-742. doi: 10.1051/m2an:2007038.  Google Scholar

[7]

H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems. 2nd ed., Springer-Verlag, Berlin, 2008.  Google Scholar

[1]

ShinJa Jeong, Mi-Young Kim. Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems. Electronic Research Archive, 2021, 29 (2) : 1991-2006. doi: 10.3934/era.2020101

[2]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[3]

Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062

[4]

Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure & Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031

[5]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297

[6]

Huan-Zhen Chen, Zhao-Jie Zhou, Hong Wang, Hong-Ying Man. An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 325-341. doi: 10.3934/dcdsb.2011.15.325

[7]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[8]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007

[9]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203

[10]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089

[11]

Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015

[12]

Md. Rabiul Haque, Takayoshi Ogawa, Ryuichi Sato. Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space. Communications on Pure & Applied Analysis, 2020, 19 (2) : 677-697. doi: 10.3934/cpaa.2020031

[13]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

[14]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319

[15]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

[16]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663

[17]

Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659

[18]

Youngmok Jeon, Eun-Jae Park. Cell boundary element methods for convection-diffusion equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 309-319. doi: 10.3934/cpaa.2006.5.309

[19]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029

[20]

M. González, J. Jansson, S. Korotov. A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems. Conference Publications, 2015, 2015 (special) : 525-532. doi: 10.3934/proc.2015.0525

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]