Friday, February 10, 2017 - 2:00pm
Beyond inf-sup: Stability estimates for multi-field variational principles by means of energetic conditions in incremental form
It is well known that mixed finite element methods have to satisfy certain criteria to provide solvability and stability. The latter criterion is, in the classical context of two-field saddle-point problems such as Stokes flow or quasi-incompressible elasticity, ensured by finite element types that satisfy the well-known inf-sup condition to ensure mesh-independent stability estimates. A number of finite element methods for novel multi-physics applications such as coupled Cahn-Hilliard-type flow in elastic media, extended phase-field models for fracture or topology optimization as well as gradient-extended plasticity models have a similar saddle-point structure. However, they correspond to a multi-field variational principle and only some of them suffer from similar instabilities. The question as to whether stability estimates are satisfied in these cases for standard discretizations and, if not, how conditions can be obtained that satisfy these estimates is particularly challenging.