# Neytcheva 2007 Feb 28

### From NA-Wiki

**Speaker:** Maya Neytcheva, Uppsala University

**Title:** Preconditioning of nonsymmetric saddle point systems with application to visco-elastic problems

**When:** The NA seminar meets om Wed Feb 28 at 15:15 in D4523

**Abstract**

This presentation is concerned with constructing efficient block-preconditioners for matrices arising from FEM approximations of various problems modelled by PDEs. In particular, we focus on matrices of saddle point form (symmetric and nonsymmetric).

The background application task is to perform numerical simulations of the so-called glacial rebound phenomenon. The problem originates from modelling the response of the solid Earth to large scale glacial advance and recession which may have provoked very large earthquakes in Northern Scandinavia. The need for such numerical simulations is due to ongoing investigations on safety assessment of radioactive waste repositories.

Within this study we use the so-called isostatic model, based on the concept that the elevation of the Earth's surface seeks a balance between the weight of lithospheric rocks and the buoyancy of asthenospheric "fluid" (nearly-molten rock). The model describes the geophysical problem in terms of a system of partial differential equations which describe the equilibrium state of a pre-stressed visco-elastic material body, subject to surface and body forces. It includes a first-order term representing the so-called advection of pre-stress, the incorporation of which has proven to be crucial for the successful modelling of the underlying processes.

The continuous setting of the problem is to solve an integro-differential equation in a large time-space domain. This problem is then discretized using a finite element method in space and a suitable discretization in time, and gives rice to the solution of a large number of linear systems with nonsymmetric matrices of saddle point form. In the purely elastic case the resulting linear systems resemble the linearized Navier-Stokes equations but is of somewhat more general form.

The presentation outlines the so-arising linear systems of equations and discusses possible preconditioning strategies, which are illustrated with numerical experiments.