The Conjugate Gradients in Subsurface Flow and Land Subsidence Modelling

Abstract

The gradient methods for the solution of large sparse sets of linear finite element equations in subsurface flow and land subsidence modeling may be extraordinarily accelerated by a suitable matrix modification. While the accelerated steepest descent turns out to converge fairly fast, the modified conjugate gradient (MCG) scheme exhibits a much faster convergence. For the range of the problem size N explored in the present paper (800≤N≤2200) the iterations required by MCG to provide very accurate results prove to be of the order of $\sqrt {\text{N}} .$

The finite element mesh and the nodal ordering need not satisfy any special property. No estimate of acceleration parameter is required. A comparison with other solvers commonly used for finite element sparse problems shows that the superiority of MCG increases with N: for N close to 1500 the experiments provide good evidence that the computer time may be reduced by a factor 5 relative to the best traditional solution techniques J Therefore at present MCG appears to be the most efficient tool for the accurate solution of the algebraic linear equations arising in finite element modeling of groundwater flow and land subsidence in realistically complex hydrogeologic settings with a non-homogeneous distribution of hydromechanical parameters, a mesh of arbitrary shape and a very large number of nodes.