SAMG for systems of equations with purely algebraic constraints
The object of this project is to extend the application areas of the SAMG solver package. SAMG already offers Uzawa variants as smoothers in multigrid cycling for saddle point problems in the Navier-Stokes area. For many important problems in the areas of mixed finite element methods, contact and obstacle problems in mechanics, geomechanics or in the area of continuum scale material design (microstructure optimization), the matrices to be solved have a similar structure, but this Uzawa smoothier cannot be used successfully because the physical background of the constraints or saddle point structure is completely different. Here SAMG has to treat certain equations as real algebraic constraints.
It is well known that many simulation codes set up their equations to be solved with such Lagrange multipliers as additional algebraic equations. Typically, however, all iterative methods have greatest difficulties to solve such equations at all - let alone efficiently. So far, direct solvers in large simulation codes have been considered as indispensable standard solvers. We want to change this by our new developments for the mentioned application classes. Therefore, new strategies and algorithms have to be developed that integrate these equations into the multigrid hierarchy in a suitable way. Also new transfer operators for AMG cycling, suitable approximations for a Schur complement as well as suitable methods must be developed, which have good smoothing properties even in the presence of algebraic equations, which do not stem directly from the discretization of partial differential equations.
Project duration: 12/2018 - 11/2021