Geodesic
Convergence of $H^1$-Galerkin mixed finite element method for parabolic problems with reduced regularity of initial data
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 17 (2014) no. 3, pp. 273-288

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We study the convergence of an $H^1$1-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming reduced regularity of the initial data. More precisely, for a spatially discrete scheme error estimates of order $\mathcal O(h^2t^{-1/2})$ for positive time are established assuming the initial function $p_0\in H^2(\Omega)\cap H_0^1(\Omega)$. Further, we use an energy technique together with a parabolic duality argument to derive error estimates of order $\mathcal O(h^2t^{-1})$ when $p_0$ is only in $H_0^1(\Omega)$. A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established. 
@article{SJVM_2014_17_3_a4,
     author = {M. Tripathy and Rajen Kumar Sinha},
     title = {Convergence of $H^1${-Galerkin} mixed finite element method for parabolic problems with reduced regularity of initial data},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {273--288},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2014_17_3_a4/}
}
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M. Tripathy; Rajen Kumar Sinha. Convergence of $H^1$-Galerkin mixed finite element method for parabolic problems with reduced regularity of initial data. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 17 (2014) no. 3, pp. 273-288. http://geodesic.mathdoc.fr/item/SJVM_2014_17_3_a4/