Developments in Preconditioned Iterative Methods with Application to Glacial Isostatic Adjustment Mo- dels

This study examines the block lower-triangular preconditioner with element-wise Schur complement as the lower diagonal block applied on matrices arising from an application in geophysics. The element-wise Schur complement is a special approximation of the exact Schur complement that can be constructed in the finite element framework. The preconditioner, the exact Schur complement and the element-wise Schur complement are analyzed mathematically and experimentally. The preconditioner is developed specifically for the glacial isostatic adjustment (GIA) model in its simplified flat Earth variant, but it is applicable to linear system of equations with matrices of saddle point form. In this work we investigate the quality of the element-wise Schur complement for symmetric indefinite matrices with positive definite pivot block and show spectral bounds that are independent of the problem size. For non-symmetric matrices we use generalized locally Toeplitz (GLT) sequences to construct a function that asymptotically describes the spectrum of the involved matrices. The theoretical results are verified by numerical experiments for the GIA model. The results show that the so-obtained preconditioned iterative method converges to the solution in constant number of iterations regardless of the problem size or parameters.