Part 1 : Using an artificial compressibility

Saddle point problems frequently appear in many mathematical and engineering applications. Most sets of partial differential equations with constraints give rise to saddle point linear systems. This is the case, for instance, when one uses mixed finite element formulations to solve fluid flows and/or elasticity problems under full incompressibility. Many direct and iterative methods have been proposed to overcome the challenges of saddle point problems due to indefiniteness and often poor spectral properties, such as the Schur complement and the Uzawa’s method. However, each method relies on the sub-block matrices properties in order to ensure stability and convergence. In the context of mixed finite element for incompressible flows using stable H(div)-L2 spaces for velocity and pressure, respectively, we have been developing iterative methods that can effectively solve a saddle point problem by introducing a small compressibility to the original matrix allowing for the static condensation of pressures. The resulting matrix is symmetric positive-definite, which allows the usage of Cholesky decomposition or CG-like iterative solvers to compute the incremental solution for the velocities unknowns. The pressure correction was shown to be proportional to the unbalanced force caused by the compressibility perturbation, and can be explicitly updated during the iterative process once the velocity increment is obtained.

Part 2 : Iterative method to solve mixed high order approximations of Darcy problem

In this work, we propose an iterative scheme to solve problems arising in the context of De Rham compatible mixed finite element approximations of Darcy equations. The strategy consists on using a Cholesky inverted matrix of constant fluxes as a preconditioner to solve the higher-order flux problem. The latter is solved iteratively by means of a conjugate gradient scheme. The preconditioner is spectrally equivalent to the original matrix and is convergent in a few iterations for any mesh size and/or polynomial order in both 2 and 3 dimensions. In addition, as internal fluxes and pressures are condensed, only boundary variables are computed.

Bio: Dr. Philippe Devloo is a distinguished professor at the State University of Campinas (UNICAMP) in the Department of Civil Engineering, Architecture, and Urbanism. He holds a Ph.D. in Aerospace Engineering and Engineering Mechanics from the University of Texas at Austin, where he studied under John Tinsley Oden. His expertise lies in numerical simulation and finite element methods, with applications in solid and fluid mechanics, specifically for the oil and aeronautic industries. He has contributed extensively to research on multiscale and multiphysics problems, hydraulic fracturing, and computational fluid dynamics. His academic and research achievements are supported by various grants and collaborations with industry partners like Petrobras and Embraer, making him a prominent figure in computational engineering.