A Least-Squares Finite Element Method for TPM
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Porous materials are distinguished by the intricate interplay between the deformation of their solid framework and the movement of the fluid within their voids. The range of applications for these materials are broad, extending from the principles of soil mechanics to the intricate details of biological tissue. Therefore, the development of precise and resilient solution methods is of paramount importance. However, challenges may emerge when the constants in the models fluctuate significantly, potentially resulting in locking when dealing with nearly incompressible solids, or in the case of low permeability materials, leading to fluctuations in pressure. To tackle these difficulties, we discuss a Least-Squares finite element method for the Biot problem, which combines exact stress approximation through constrained minimization and robustness with respect to the incompressible limit. Then a framework is presented with allows the a posteriori analysis for a class of nonlinear problems for the Least-Squares functional. We then use this framework to deal with the TPM(Theory of Porous Media) model. Finally, numerical examples are presented.
