Saddle-Point Problems in Computational and Fracture Mechanics: modeling and efficient solution

Saddle-point problems naturally emerge in various coupled and constrained problems within computational mechanics, including incompressible elasticity, poroelasticity, and thermoelasticity. Multigrid solvers stand out as among the fastest methods for solving differential problems discretized using finite element or finite difference methods. These solvers are characterized by an optimal complexity and optimal convergence rates independent of the number of unknowns, particularly for unconstrained and uncoupled elliptic problems characterized by high solution regularity.

In this talk, we aim to address two primary objectives. Firstly, we will thoroughly investigate the performance evaluation of multigrid solvers for nonlinear saddle-point problems encountered in computational mechanics. We will explore effective combinations of these solvers with both monolithic and staggered schemes for addressing coupled problems in computational mechanics. Additionally, we will examine instances of failure of their convergence, often arising from erroneous or impractical choices in the computational meshes of the underlying finite element discretization. Furthermore, we will discuss how appropriate stabilizations can mitigate these issues and enhance the convergence of multigrid methods.

Secondly, we will focus on mechanical problems within fracture mechanics. The presence of fractures, particularly intricate networks, poses significant challenges in generating suitable meshes. Moreover, fractures diminish the regularity of the solution, impacting the convergence of both the underlying finite element discretization and multigrid solvers. We will explore modeling techniques aimed at reformulating problems with thin heterogeneities into interface problems. These novel formulations involve slight modifications of the original problem, characterized by both non-standard interface conditions and a higher regularity of the solution.