Geodesic
Two multigrid iterative algorithms for a~discrete analogue of the biharmonic equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 3, pp. 213-228

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A standard scheme of the finite element method with the use of bicubic elements on a rectangular quasiuniform grid is considered as applied to the two-dimensional Dirichlet problem for the biharmonic equation in a rectangle. To solve this scheme, two multigrid algorithms are treated on a sequence of embedded rectangular grids: a full multigrid with $V$-cycle and a simpler cascadic algorithm. The presence of angular points of a rectangle results in deficiency of solution smoothness which complicates substantiation of convergence of the algorithm as compared to a smooth case. At the same time, a number of arithmetical operations remains almost optimal for the cascadic algorithm and optimal for $V$-cycles. 
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     author = {L. V. Gilyova and V. V. Shaidurov},
     title = {Two multigrid iterative algorithms for a~discrete analogue of the biharmonic equation},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {213--228},
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     url = {http://geodesic.mathdoc.fr/item/SJVM_2004_7_3_a3/}
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L. V. Gilyova; V. V. Shaidurov. Two multigrid iterative algorithms for a~discrete analogue of the biharmonic equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 7 (2004) no. 3, pp. 213-228. http://geodesic.mathdoc.fr/item/SJVM_2004_7_3_a3/