Galerkin approximations for the linear parabolic equation with the third boundary condition
István Faragó; Sergey Korotov; Pekka Neittaanmäki
Applications of Mathematics (2003)
- Volume: 48, Issue: 2, page 111-128
- ISSN: 0862-7940
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topFaragó, István, Korotov, Sergey, and Neittaanmäki, Pekka. "Galerkin approximations for the linear parabolic equation with the third boundary condition." Applications of Mathematics 48.2 (2003): 111-128. <http://eudml.org/doc/33139>.
@article{Faragó2003,
abstract = {We solve a linear parabolic equation in $\mathbb \{R\}^d$, $d \ge 1,$ with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the $\theta $-method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.},
author = {Faragó, István, Korotov, Sergey, Neittaanmäki, Pekka},
journal = {Applications of Mathematics},
keywords = {linear parabolic equation; third boundary condition; finite element method; semidiscretization; fully discretized scheme; elliptic projection; finite element method; semidiscretization; fully discretized scheme; elliptic projection},
language = {eng},
number = {2},
pages = {111-128},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Galerkin approximations for the linear parabolic equation with the third boundary condition},
url = {http://eudml.org/doc/33139},
volume = {48},
year = {2003},
}
TY - JOUR
AU - Faragó, István
AU - Korotov, Sergey
AU - Neittaanmäki, Pekka
TI - Galerkin approximations for the linear parabolic equation with the third boundary condition
JO - Applications of Mathematics
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 2
SP - 111
EP - 128
AB - We solve a linear parabolic equation in $\mathbb {R}^d$, $d \ge 1,$ with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the $\theta $-method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.
LA - eng
KW - linear parabolic equation; third boundary condition; finite element method; semidiscretization; fully discretized scheme; elliptic projection; finite element method; semidiscretization; fully discretized scheme; elliptic projection
UR - http://eudml.org/doc/33139
ER -
References
top- On a conservative, high-order accurate finite element scheme for the “parabolic” equation, Comput. Acoustics 1 (1989), 17–26. (1989) MR1095058
- Introduction to Matrix Analysis, SIAM, Philadelphia, 1997. (1997) Zbl0872.15003MR1455129
- The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, Inc. A, 1994. (1994) MR1278258
- 10.1137/0707048, SIAM J. Numer. Anal. 7 (1970), 575–626. (1970) MR0277126DOI10.1137/0707048
- The numerical solution of waterflooding problems in petroleum engineering by variational methods, Studies in Numerical Analysis 2, SIAM, Philadelphia, 1970. (1970) MR0269141
- The application of variational methods to waterflooding problems, J. Canad. Petroleum Tech. 8 (1969), 79–85. (1969)
- 10.1090/S0025-5718-1978-0495012-2, Math. Comp. 32 (1978), 345–362. (1978) MR0495012DOI10.1090/S0025-5718-1978-0495012-2
- Finite element analysis for the heat conduction equation with the third boundary condition, Annales Univ. Sci. Budapest 41 (1998), 183–195. (1998) MR1691927
- 10.1007/BF01402523, Numer. Math. 19 (1972), 127–135. (1972) MR0311122DOI10.1007/BF01402523
- Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications, Kluwer Academic Publishers, 1996. (1996) MR1431889
- Calculation of the 3D temperature field of synchronous and of induction machines by the finite element method, Elektrotechn. obzor 80 (1991), 78–84. (Czech) (1991)
- 10.1093/imamat/19.1.119, J. Inst. Math. Appl. 19 (1977), 119–125. (1977) Zbl0346.65040MR0440926DOI10.1093/imamat/19.1.119
- 10.2118/1877-PA, Soc. Petroleum Engrg. J. 8 (1968), 293–303. (1968) DOI10.2118/1877-PA
- Error bounds for semi-discrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics, Numerical Solution of Field Problems in Continuum Physics, AMS, Providence, 1970, pp. 74–94. (1970) MR0266452
- Theory of Difference Schemes, Nauka, Moscow, 1977. (Russian) (1977) Zbl0368.65031MR0483271
- Finite difference method for parabolic problems with nonsmooth initial data, Report of Oxford Univ. Comp. Lab. 86/22 (1987). (1987)
- An Analysis of the Finite Element Method, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1973. (1973) MR0443377
- Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 1997. (1997) MR1479170
- Functional Analysis and Approximation Theory in Numerical Analysis, SIAM, Philadelphia, 1971. (1971) Zbl0226.65064MR0310504
- 10.1137/0710062, SIAM J. Numer. Anal. 10 (1973), 723–759. (1973) MR0351124DOI10.1137/0710062
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