Volume 10, Issue 3
Error Analysis of Linearized Semi-Implicit Galerkin Finite Element Methods for Nonlinear Parabolic Equations

Int. J. Numer. Anal. Mod., 10 (2013), pp. 622-633.

Published online: 2013-10

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This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without any time-step restriction. Theoretical analysis is based on a new splitting of error function and precise analysis of a corresponding time-discrete system. The method used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations for which previous works often require certain restriction on the time-step size $\tau$.

65N12, 65N30, 35K61

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@Article{IJNAM-10-622, author = {}, title = {Error Analysis of Linearized Semi-Implicit Galerkin Finite Element Methods for Nonlinear Parabolic Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {3}, pages = {622--633}, abstract = {

This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without any time-step restriction. Theoretical analysis is based on a new splitting of error function and precise analysis of a corresponding time-discrete system. The method used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations for which previous works often require certain restriction on the time-step size $\tau$.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/586.html} }
TY - JOUR T1 - Error Analysis of Linearized Semi-Implicit Galerkin Finite Element Methods for Nonlinear Parabolic Equations JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 622 EP - 633 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/586.html KW - Nonlinear parabolic system, unconditionally optimal error estimate, linearized semi-implicit scheme, Galerkin method. AB -

This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without any time-step restriction. Theoretical analysis is based on a new splitting of error function and precise analysis of a corresponding time-discrete system. The method used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations for which previous works often require certain restriction on the time-step size $\tau$.

B. Li & W. Sun. (1970). Error Analysis of Linearized Semi-Implicit Galerkin Finite Element Methods for Nonlinear Parabolic Equations. International Journal of Numerical Analysis and Modeling. 10 (3). 622-633. doi:
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