Browse > Article
http://dx.doi.org/10.7858/eamj.2022.018

A HIGHER ORDER SPLIT LEAST-SQUARES CHARACTERISTIC MIXED ELEMENT METHOD FOR SOBOLEV EQUATIONS  

Ohm, Mi Ray (Division of Mechatronics Engineering Dongseo University)
Shin, Jun Yong (Department of Applied Mathematics Pukyong National University)
Publication Information
East Asian mathematical journal / v.38, no.3, 2022 , pp. 293-319 More about this Journal
Abstract
In this paper, we introduce a higher order split least-squares characteristic mixed element scheme for Sobolev equations. First, we use a characteristic mixed element method to manipulate both convection term and time derivative term efficiently and obtain the system of equations in the primal unknown and the flux unknown. Second, we define a least-squares minimization problem and a least-squares characteristic mixed element scheme. Finally, we obtain a split least-squares characteristic mixed element scheme for the given problem whose system is uncoupled in the unknowns. We establish the convergence results for the primal unknown and the flux unknown with the second order in a time increment.
Keywords
Sobolev equations; a split least-squares method; characteristic mixed element method;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 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
2 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
3 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
4 D. M. Shi, On the initial boundary value problem of the nonlinear equation of the migration of the moisture in soil, Acta math. Appl. Sinica 13 (1990), 31-38.
5 D. P. Yang, Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems, Math. Comput. 69 (2000), 929-963.   DOI
6 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.   DOI
7 M. R. Ohm, H. Y. Lee and J. Y. Shin, L2 -error analysis of discontinuous Galerkin approximations for nonlinear Sobolev equations, J. Japanese Indus. Appl. Math. 30(1) (2013), 91-110.   DOI
8 H. Gu, Characteristic finite element methods for nonlinear Sobolev equations, Applied Math. Comput. 102 (1999), 51-62.   DOI
9 K. Boukir, Y. Maday and B. Metivet, A high-order characteristics/finite element method for the incompressible navier-stokes equations, Inter. Jour. Numer. Methods in Fluids. 25 (1997), 1421-1454.   DOI
10 T. Arbogast and M. Wheeler, A characteristics-mixed finite element method for advection-dominated transport problem, SIAM J. Numer. Anal. 32(2) (1995), 404-424.   DOI
11 Z. Chen, Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems, Comput. Methods. Appl. Mech. Engrg. 191 (2002), 2509-2538.   DOI
12 M. R. Ohm and H. Y. Lee, L2-error analysis of fully discrete discontinuous Galerkin approximations for nonlinear Sobolev equations, Bull. Korean. Math. Soc. 48(5) (2011), 897-915.   DOI
13 Z. Chen, R. Ewing, Q. Jiang and A. Spagnuolo, Error analysis for characteristics-based methods for degenerate parabolic problems, SIAM J. Numer. Anal. 40(4) (2002), 1491-1515.   DOI
14 C. Dawson, T. Russell and M. Wheeler, Some improved error estimates for the modified method of characteristics, SIAM J. Numer. Anal. 26(6) (1989), 1487-1512.   DOI
15 F. Gao and H. Rui, A split least-squares characteristic mixed finite element method for Sobolev equations with convection term, Math. Comput. Simulation 80 (2009), 341-351.   DOI
16 L. Guo and H. Z. Chen, H1-Galerkin mixed finite element method for the Sobolev equation, J. Sys. Sci. 26 (2006), 301-314.
17 X. Long and C. Chen, Implicit-Explicit multistep characteristic finite element methods for nonlinear convection-diffusion equations, Numer. Methods Parial Differential Eq. 23 (2007), 1321-1342.   DOI
18 M. R. Ohm and J. Y. Shin, A split least-squares characteristics mixed finite element method for the convection dominated Sobolev equations, J. Appl. Math. Informatics. 34(1) (2016), 19-34.   DOI
19 P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Proc. Conf. on Mathemaical Aspects of Finite Element Methods, Lecture Notes in Math., Vol. 606, Springer-Verlag, Berlin, 1977, 292-315.
20 R. E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev equations, SIAM J. Numer. Anal. 15 (1978), 1125-1150.   DOI
21 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
22 T. W. Ting, A cooling process according to two-temperature theory of heat conduction, J. Math. Anal. Appl. 45 (1974), 23-31.   DOI
23 D. P. Yang, Some least-squares Galerkin procedures for first-order time-dependent convection-diffusion system, Comput. Methods Appl. Mech. Eng. 108 (1999), 81-95.   DOI
24 J. Zhang amd H. Guo, A split least-squares characteristic mixed element method for nonlinear nonstationary convection-diffusion problem, Int. J. Comput. Math. 89 (2012), 932-943.   DOI
25 M. R. Ohm and J. Y. Shin, A split least-squares characteristics mixed element method for Sobolev equations with a convection term, East Asian Math. J. 35(5) (2019), 569-587.   DOI
26 H. Guo, A remark on split least-squares mixed element procedures for pseudo-parabolic equations, Applied Math. Comput. 217 (2011), 4682-4690.   DOI