An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.
Keywords: finite element method, local projection stabilization, crosswind diffusion, convection-diffusion-reaction equation, well posedness, time dependent problem, stability, error estimates
@article{M2AN_2013__47_5_1335_0, author = {Barrenechea, Gabriel R. and John, Volker and Knobloch, Petr}, title = {A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1335--1366}, publisher = {EDP-Sciences}, volume = {47}, number = {5}, year = {2013}, doi = {10.1051/m2an/2013071}, mrnumber = {3100766}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2013071/} }
TY - JOUR AU - Barrenechea, Gabriel R. AU - John, Volker AU - Knobloch, Petr TI - A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1335 EP - 1366 VL - 47 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013071/ DO - 10.1051/m2an/2013071 LA - en ID - M2AN_2013__47_5_1335_0 ER -
%0 Journal Article %A Barrenechea, Gabriel R. %A John, Volker %A Knobloch, Petr %T A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1335-1366 %V 47 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013071/ %R 10.1051/m2an/2013071 %G en %F M2AN_2013__47_5_1335_0
Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr. A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 5, pp. 1335-1366. doi : 10.1051/m2an/2013071. http://www.numdam.org/articles/10.1051/m2an/2013071/
[1] An assessment of discretizations for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Engrg. 200 (2011) 3395-3409. | MR | Zbl
, , , , , and ,[2] A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001) 173-199. | MR | Zbl
and ,[3] A two-level stabilization scheme for the Navier-Stokes equations, Proc. of ENUMATH 2003, Numerical Mathematics and Advanced Applications, edited by M. Feistauer, V. Dolejıš´, P. Knobloch and K. Najzar. Springer-Verlag, Berlin (2004) 123-130. | MR | Zbl
and ,[4] Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43 (2006) 2544-2566. | MR | Zbl
and ,[5] Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853-866. | MR | Zbl
, , and ,[6] Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199-259. | MR | Zbl
and ,[7] Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence. Math. Comput. 74 (2005) 1637-1652. | MR | Zbl
and ,[8] Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 198 (2009) 2508-2519. | MR | Zbl
and ,[9] Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437-1453. | MR | Zbl
and ,[10] The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl
,[11] A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation. Comput. Methods Appl. Mech. Engrg. 110 (1993) 325-342. | MR | Zbl
,[12] Theory and Practice of Finite Elements. Springer-Verlag, New York (2004). | MR | Zbl
and ,[13] Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg. 95 (1992) 253-276. | MR | Zbl
, and ,[14] On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation. Comput. Methods Appl. Mech. Engrg. 190 (2000) 1785-1800. | MR | Zbl
and ,[15] Stabilization by local projection for convection-diffusion and incompressible flow problems. J. Sci. Comput. 43 (2010) 326-342. | MR | Zbl
and ,[16] A new finite element formulation for computational fluid dynamics. VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg. 73 (1989) 173-189. | MR | Zbl
, and ,[17] 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. | MR | Zbl
and ,[18] On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part II - Analysis for P1 and Q1 finite elements. Comput. Methods Appl. Mech. Engrg. 197 (2008) 1997-2014. | MR | Zbl
and ,[19] A posteriori optimization of parameters in stabilized methods for convection-diffusion problems - Part I. Comput. Methods Appl. Mech. Engrg. 200 (2011) 2916-2929. | MR | Zbl
, and ,[20] Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78 (1997) 165-188. | MR | Zbl
, and ,[21] Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Engrg. Sci. 64 (2009) 733-741.
, , , , and ,[22] Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations. SIAM J. Numer. Anal. 49 (2011) 1149-1176. | MR | Zbl
and ,[23] Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion. Comput. Methods Appl. Mech. Engrg. 198 (2008) 475-494. | MR | Zbl
and ,[24] A generalization of the local projection stabilization for convection-diffusion-reaction equations. SIAM J. Numer. Anal. 48 (2010) 659-680. | MR | Zbl
,[25] Local projection method for convection-diffusion-reaction problems with projection spaces defined on overlapping sets. Proc. of ENUMATH 2009, Numerical Mathematics and Advanced Applications, edited by G. Kreiss, P. Lötstedt, A. M?lqvist and M. Neytcheva. Springer-Verlag, Berlin (2010) 497-505. | Zbl
,[26] Local projection stabilization for advection-diffusion-reaction problems: One-level vs. two-level approach. Appl. Numer. Math. 59 (2009) 2891-2907. | MR | Zbl
and ,[27] Stabilized finite element methods with shock capturing for advection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2997-3013. | MR | Zbl
, and ,[28] New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them. Tr. Mat. Inst. Steklova 102 (1967) 85-104. | MR | Zbl
,[29] G. Lube and G. Rapin, residual-based stabilized higher-order FEM for advection-dominated problems. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4124-4138. | MR | Zbl
[30] A unified convergence analysis for local projection stabilizations applied to the Oseen problem. Math. Model. Numer. Anal. 41 (2007) 713-742. | EuDML | Numdam | MR | Zbl
, and ,[31] Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems, 2nd ed. Springer-Verlag, Berlin (2008). | MR | Zbl
, and ,[32] Navier-Stokes Equations. Theory and Numerical Analysis North-Holland, Amsterdam (1977). | MR | Zbl
,Cited by Sources: