Volume 1, Issue 2
An $H^1$-Galerkin Mixed Method for Second Order Hyperbolic Equations

Int. J. Numer. Anal. Mod., 1 (2004), pp. 111-130.

Published online: 2004-01

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• Abstract

An $H^1$-Galerkin mixed finite element method is discussed for a class of second order hyperbolic problems. It is proved that the Galerkin approximations have the same rates of convergence as in the classical mixed method, but without LBB stability condition and quasi-uniformity requirement on the finite element mesh. Compared to the results proved for one space variable, the $L^∞(L^2)$-estimate of the stress is not optimal with respect to the approximation property for the problems in two and three space dimensions. It is further noted that if the Raviart- Thomas spaces are used for approximating the stress, then optimal estimate in $L^∞(L^2)$-norm is achieved using the new formulation. Finally, without restricting the approximating spaces for the stress, a modification of the method is proposed and analyzed. This confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed procedure under an extra regularity assumption on the exact solution.

• Keywords

Second order wave equation, LBB condition, $H^1$ Galerkin mixed finite element method, semidiscrete scheme, completely discrete method, optimal error estimates.

65M60, 65M15, 65M12

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@Article{IJNAM-1-111, author = {A. K. and Pani and and 21293 and and A. K. Pani and R. K. and Sinha and and 21294 and and R. K. Sinha and A. K. and Otta and and 21295 and and A. K. Otta}, title = {An $H^1$-Galerkin Mixed Method for Second Order Hyperbolic Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2004}, volume = {1}, number = {2}, pages = {111--130}, abstract = {

An $H^1$-Galerkin mixed finite element method is discussed for a class of second order hyperbolic problems. It is proved that the Galerkin approximations have the same rates of convergence as in the classical mixed method, but without LBB stability condition and quasi-uniformity requirement on the finite element mesh. Compared to the results proved for one space variable, the $L^∞(L^2)$-estimate of the stress is not optimal with respect to the approximation property for the problems in two and three space dimensions. It is further noted that if the Raviart- Thomas spaces are used for approximating the stress, then optimal estimate in $L^∞(L^2)$-norm is achieved using the new formulation. Finally, without restricting the approximating spaces for the stress, a modification of the method is proposed and analyzed. This confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed procedure under an extra regularity assumption on the exact solution.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/969.html} }
TY - JOUR T1 - An $H^1$-Galerkin Mixed Method for Second Order Hyperbolic Equations AU - Pani , A. K. AU - Sinha , R. K. AU - Otta , A. K. JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 111 EP - 130 PY - 2004 DA - 2004/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/969.html KW - Second order wave equation, LBB condition, $H^1$ Galerkin mixed finite element method, semidiscrete scheme, completely discrete method, optimal error estimates. AB -

An $H^1$-Galerkin mixed finite element method is discussed for a class of second order hyperbolic problems. It is proved that the Galerkin approximations have the same rates of convergence as in the classical mixed method, but without LBB stability condition and quasi-uniformity requirement on the finite element mesh. Compared to the results proved for one space variable, the $L^∞(L^2)$-estimate of the stress is not optimal with respect to the approximation property for the problems in two and three space dimensions. It is further noted that if the Raviart- Thomas spaces are used for approximating the stress, then optimal estimate in $L^∞(L^2)$-norm is achieved using the new formulation. Finally, without restricting the approximating spaces for the stress, a modification of the method is proposed and analyzed. This confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed procedure under an extra regularity assumption on the exact solution.

A. K. Pani, R. K. Sinha & A. K. Otta. (1970). An $H^1$-Galerkin Mixed Method for Second Order Hyperbolic Equations. International Journal of Numerical Analysis and Modeling. 1 (2). 111-130. doi:
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