Parallel preconditioning and hybridization techniques for a stress-velocity formulation for highly variable parameters Stokes problem

Abstract:

The usual velocity-pressure formulation for Stokes problem has been deeply studied and a lot of work has been devoted in order to achieve high performance and robust solvers, in particular highly variable parameters seems to be the more challenging difficulty. Obviously this kind of problem is very interesting from the geopyshical point of view, since the Stokes equations are used to model various geopydynamical processes: like the dynamic of lithosphere and asthenosphere.

We present a novel stress-velocity formulation, that can be intepreted as a real mixed formulation for the Stokes problem. Even if this formulation looks technically more challenging with respect to the usual velocity-pressure formulation, it seems to be more suitable for solving problem with highly variable parameters. From the physical point this formulation enforces the continuity of the normal component of stress, on the other hand the velocity-pressure formulation ensures the continuity of velocity, for this reason the can interpret the first as dynamic formulation and the latter as kinematic one.

Due to the symmetry of the stress tensor also the discretization of this problem is challenging and if this constraint is not handled properly then the problem can be not well-posed or, otherwise, a huge number of degrees of freedom per cell is needed. In order to circumvent this problems the symmetry is enforced only weakly, as proposed by other authors.

We show how an incorrect choice of the finite element spaces for the velocity-pressure formulation leads to less accurate results even if the approximate problem is well-posed, on the other hand the finite element formulation of stress-velocity formulation leads to the correct results in a natural way.

We present an embarrassingly parallel preconditioning techique which is very robust with respect to parameters variations, in particular for discontinuous parameters. Moreover the problem can be hybridized in order to reduce the number of degrees of freedom: in particular the stress-velocity formulation can be reduced to another saddle-point problem where the unknowns are the velocity, defined only on the elements' boundary, and another auxiliary variable, for which only one degree of freedom per cell is needed regardless of approximation order.