Browse > Article
http://dx.doi.org/10.14317/jami.2018.257

AN EXTRAPOLATED CRANK-NICOLSON CHARACTERISTIC FINITE ELEMENT METHOD FOR NONLINEAR SOBOLEV EQUATIONS  

OHM, MI RAY (Division of Mechatronics Engineering, Dongseo University)
SHIN, JUN YONG (Department of Applied Mathematics, College of Natural Sciences, Pukyong National University)
Publication Information
Journal of applied mathematics & informatics / v.36, no.3_4, 2018 , pp. 257-270 More about this Journal
Abstract
An extrapolated Crank-Nicolson characteristic finite element method is introduced for approximate solutions of nonlinear Sobolev equations with a convection term. And we obtain the higher order of convergence for approximate solutions in the temporal and the spatial directions with respect to $L^2$ norm.
Keywords
nonlinear Sobolev equation; an extrapolated Crank-Nicolson characteristic finite element method; higher order of convergence;
Citations & Related Records
Times Cited By KSCI : 4  (Citation Analysis)
연도 인용수 순위
1 M.R. Ohm, H.Y. Lee, and J.Y. Shin, $L^2$-error analysis of discontinuous Galerkin approximations for nonlinear Sobolev equations, Jpn. J. Ind. Appl. Math. 30(1) (2013) 91-110.   DOI
2 M.R. Ohm and J.Y. Shin, A Crank-Nicolson characteristic finite element method for Sobolev equations, East Asian Math. J. 32(5) (2016) 729-744.   DOI
3 M.R. Ohm and J.Y. Shin, A Crank-Nicolson characteristic finite element method for nonlinear Sobolev equations, East Asian Math. J. 33(3) (2017) 295-308.   DOI
4 M.R. Ohm and J.Y. Shin, An extrapolated Crank-Nicolson characteristic finite element method for Sobolev equations, Bull. Korean Math. Soc. 54(4) (2017) 1409-1419.   DOI
5 A. Pehlivanov, G.F. Carey, and D. Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J. Numer. Anal. 31 (1994) 1368-1377.   DOI
6 H.X. Rui, S. Kim, and S.D. Kim, A remark on least-squares mixed element methods for reaction-diffusion problems, J. Comput. Appl. Math. 202 (2007) 203-236.   DOI
7 D. Shi, Q. Tang, and W. Gong, A lower order characteristic-nonconforming finite element method for nonlinear Sobolev equation with convection-dominated term, Math. Comput. Simulation 114 (2015) 25-36.   DOI
8 D.M. Shi, The initial-boundary value problem for a nonlinear equation of migration of moisture in soil, Acta Math. Appl. Sinica 13 (1990) 31-38.
9 T. Sun and D. Yang, A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations, Appl. Math. Comput. 200 (2008) 147-159.
10 T. Sun and D. Yang, Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations, Numer. Methods Partial Differential Equations 24(3) (2008) 879-896.   DOI
11 T.W. Ting, A cooling process according to two-temperature theory of heat conduction, J. Math. Anal. Appl. 45 (1974) 23-31.   DOI
12 D.P. Yang, Some least-squares Galerkin procedures for first-order time-dependent convection-diffusion system, Comput. Methods Appl. Mech. Engrg. 108 (1999) 81-95.
13 D.P. Yang, Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems, Math. Comp. 69 (2000) 929-963.
14 R.E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal. 15 (1978) 1125-1150.   DOI
15 D.N. Arnold, J.Jr. Douglas, and V. Thomee, Superconvergence of a finite approximation to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981) 53-63.   DOI
16 G.I. Barenblatt, I.P. Zheltov, and I.N. Kochian, Basic conception in the theory of seepage of homogenous liquids in fissured rocks, J. Appl. Math. Mech. 24 (1960) 1286-1309.   DOI
17 R.W. Carroll and R.E. Showalter, Singular and degenerate Cauchy problems (Mathematics in Sciences and Engineering, Vol. 127), Academic Press, New York, 1976.
18 P.L. Davis, A quasilinear parabolic and related third order problem, J. Math. Anal. Appl. 49 (1970) 327-335.
19 J. Douglas and T.F. Russell Jr., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristic with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982) 871-885.   DOI
20 H. Gu, Characteristic finite element methods for nonlinear Sobolev equations, Appl. Math. Comput. 102 (1999) 51-62.
21 L. Guo and H.Z. Chen, $H^1$-Galerkin mixed finite element method for the Sobolev equation, J. Sys. Sci. 26 (2006) 301-314.
22 M.T. Nakao, Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension, Numer. Math. 47 (1985) 139-157.   DOI
23 H. Guo and H.X. Rui, Least-squares Galerkin mixed finite element method for the Sobolev equation, Acta Math. Appl. Sinica 29 (2006) 609-618.
24 Y. Lin, Galerkin methods for nonlinear Sobolev equations, Aequationes Math. 40 (1990) 54-66.   DOI
25 Y. Lin and T. Zhang, Finite element methods for nonlinear Sobolev equations with non-linear boundary conditions, J. Math. Anal. Appl. 165 (1992) 180-191.   DOI
26 M.R. Ohm and H.Y. Lee, $L^2$-error analysis of fully discrete discontinuous Galerkin approximations for nonlinear Sobolev equations, Bull. Korean. Math. Soc. 48(5) (2011) 897- 915.   DOI