Geodesic
Solving parabolic equations on locally refined grids
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 11, pp. 2031-2043

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An implicit finite difference scheme for solving the heat conduction equation on locally refined grids in a rectangular domain is considered. To solve the resulting system of equations, the conjugate gradient method with preconditioning is used. This method is a variant of the incomplete Cholesky decomposition or modified incomplete Cholesky decomposition. A modification of the computation of the preconditioning matrix for the variant of the incomplete Cholesky-conjugate gradient method for the case of the numerical solution of heat conduction equations with a rapidly varying thermal conductivity coefficient is proposed. Variants of the above-mentioned method designed for use on parallel computer systems with MIMD architecture are proposed. The solution of model problems on a moderate number of processors is used to examine the rate of convergence and the efficiency of the proposed methods. 
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     author = {O. Yu. Milyukova and V. F. Tishkin},
     title = {Solving parabolic equations on locally refined grids},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {2031--2043},
     publisher = {mathdoc},
     volume = {45},
     number = {11},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_11_a10/}
}
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O. Yu. Milyukova; V. F. Tishkin. Solving parabolic equations on locally refined grids. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 11, pp. 2031-2043. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_11_a10/