An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients

Abstract

Knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs. A mathematical model for the prediction of fluid pressures is given by a time-dependent diffusion equation. Application of the finite element method leads to a system of linear equations. A complication is that the underground consists of layers with very large differences in permeability. This implies that the symmetric and positive definite coefficient matrix has a very large condition number. Bad convergence behavior of the CG method has been observed; moreover, a classical termination criterion is not valid in this problem. After diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability. For the IC preconditioner the same behavior is observed. To annihilate the effect of the extreme eigenvalues a deflated CG method is used. The convergence rate improves considerably and the termination criterion becomes again reliable. Finally a cheap approximation of the eigenvectors is proposed.

Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems

“…This makes the blocks of the discretization matrix that correspond to the sandwiched sandlayers almost singular. This observation is formulated in terms of the following theorem, which is proven by Vuik et al [17]:…”

“…It gives an upper bound for the error for the ||.|| 2 -norm. Vuik et al [17] observe that the number of small eigenvalues (order 10 −7 ) of A is equal to the number of gridnodes in the low mobility layer (σ = 10 −7 ) plus the number of high permeability layers that have a low permeability layer on top. The conjugate gradient method converges to the exact solution only once all small eigenvalues have been 'discovered'.…”

“…Furthermore, deflation is applied to get rid of remaining extremely small eigenvalues that delay convergence. Vuik et al [17] proposed a scheme based on physical deflation in which the deflation vectors are continuous and satisfy the original partial differential equation on a subdomain. A different variant involves the so-called algebraic deflation with discontinuous deflation vectors.…”

“…For references related to the Deflated ICCG method we refer to the overview given in [17] and [16]. The DICCG method has already been successfully used for complicated magnetic field simulations [2].…”

“…In our previous work we always use non-overlapping deflation vectors. In [17,18] the interface vertices are only elements of the high permeability subdomains, whereas in [4] no interface vertices occur due to a cell centered discretization. The topic of this paper is: how to choose the value of the deflation vectors at interface points in order to obtain an efficient, robust and parallelizable black-box deflation method.…”