Volume 13, Issue 3
Multidomain Legendre-Galerkin Least-Squares Method for Linear Differential Equations with Variable Coefficients

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 665-688.

Published online: 2020-03

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• Abstract

The multidomain Legendre-Galerkin least-squares method is developed for solving linear differential problems with variable coefficients. By introducing a flux, the original differential equation is rewritten into an equivalent first-order system, and the Legendre Galerkin is applied to the discrete form of the corresponding least squares function. The proposed scheme is based on the Legendre-Galerkin method, and the Legendre/Chebyshev-Gauss-Lobatto collocation method is used to deal with the variable coefficients and the right hand side terms. The coercivity and continuity of the method are proved and the optimal error estimate in $H^1$-norm is obtained. Numerical examples are given to validate the efficiency and spectral accuracy of our scheme. Our scheme is also applied to the numerical solutions of the parabolic problems with discontinuous coefficients and the two-dimensional elliptic problems with piecewise constant coefficients, respectively.

35F25, 65M70, 65L06

yonghui1676@163.com (Yonghui Qin)

yanpingchen@scnu.edu.cn (Yanping Chen)

huangyq@xtu.edu.cn (Yunqing Huang)

hpma@shu.edu.cn (Heping Ma)

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@Article{NMTMA-13-665, author = {Qin , YonghuiChen , YanpingHuang , Yunqing and Ma , Heping}, title = {Multidomain Legendre-Galerkin Least-Squares Method for Linear Differential Equations with Variable Coefficients}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {3}, pages = {665--688}, abstract = {

The multidomain Legendre-Galerkin least-squares method is developed for solving linear differential problems with variable coefficients. By introducing a flux, the original differential equation is rewritten into an equivalent first-order system, and the Legendre Galerkin is applied to the discrete form of the corresponding least squares function. The proposed scheme is based on the Legendre-Galerkin method, and the Legendre/Chebyshev-Gauss-Lobatto collocation method is used to deal with the variable coefficients and the right hand side terms. The coercivity and continuity of the method are proved and the optimal error estimate in $H^1$-norm is obtained. Numerical examples are given to validate the efficiency and spectral accuracy of our scheme. Our scheme is also applied to the numerical solutions of the parabolic problems with discontinuous coefficients and the two-dimensional elliptic problems with piecewise constant coefficients, respectively.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0015}, url = {http://global-sci.org/intro/article_detail/nmtma/15780.html} }
TY - JOUR T1 - Multidomain Legendre-Galerkin Least-Squares Method for Linear Differential Equations with Variable Coefficients AU - Qin , Yonghui AU - Chen , Yanping AU - Huang , Yunqing AU - Ma , Heping JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 665 EP - 688 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0015 UR - https://global-sci.org/intro/article_detail/nmtma/15780.html KW - Variable coefficient, Legendre Galerkin, Legendre/Chebyshev-Gauss-Lobatto, least squares. AB -

The multidomain Legendre-Galerkin least-squares method is developed for solving linear differential problems with variable coefficients. By introducing a flux, the original differential equation is rewritten into an equivalent first-order system, and the Legendre Galerkin is applied to the discrete form of the corresponding least squares function. The proposed scheme is based on the Legendre-Galerkin method, and the Legendre/Chebyshev-Gauss-Lobatto collocation method is used to deal with the variable coefficients and the right hand side terms. The coercivity and continuity of the method are proved and the optimal error estimate in $H^1$-norm is obtained. Numerical examples are given to validate the efficiency and spectral accuracy of our scheme. Our scheme is also applied to the numerical solutions of the parabolic problems with discontinuous coefficients and the two-dimensional elliptic problems with piecewise constant coefficients, respectively.

Yonghui Qin, Yanping Chen, Yunqing Huang & Heping Ma. (2020). Multidomain Legendre-Galerkin Least-Squares Method for Linear Differential Equations with Variable Coefficients. Numerical Mathematics: Theory, Methods and Applications. 13 (3). 665-688. doi:10.4208/nmtma.OA-2019-0015
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