An iterated tangential filtering decomposition

Large linear systems arising from the discretization of partial differential equations with finite differences or finite elements on structured grids in dimension d(d ⩾ 3) require efficient preconditioners. For a symmetric and positive definite (SPD) matrix, we propose a SPD block LDLT preconditioner whose factorized form requires a smaller amount of memory than the original matrix. Moreover, the computing time for the preconditioner solves is linear with respect to the number of unknowns. The preconditioner is built in d stages: in a first stage, we use the tangential filtering decomposition of Wittum et al. and obtain a preconditioner which remains rather difficult to factorize. Then, in a second stage, we apply tangential filtering decomposition recursively to the diagonal blocks of this first preconditioner. The final stage consists of factorizing exactly the blocks corresponding to one dimensional problems. Such preconditioners can also be computed adaptively and combined in a multiplicative way. A generic programming implementation is discussed and numerical tests are presented, in particular for problems with highly heterogeneous media. Copyright © 2003 John Wiley & Sons, Ltd.