On the optimization of the fixed-stress splitting for Biot's equations
For researchers solving poromechanics problems, this provides a principled way to choose the tuning parameter, improving convergence of the fixed-stress method.
The authors optimize the stabilization parameter in the fixed-stress splitting scheme for Biot's equations, showing it depends on fluid flow properties. Numerical tests confirm the theoretical predictions.
In this work we are interested in effectively solving the quasi-static, linear Biot model for poromechanics. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well-known that the convergence of the method is strongly dependent on the applied stabilization/tuning parameter. In this work, we propose a new approach to optimize this parameter. We show theoretically that it depends also on the fluid flow properties and not only on the mechanics properties and the coupling coefficient. The type of analysis presented in this paper is not restricted to a particular spatial discretization. We only require it to be inf-sup stable. The convergence proof applies also to low-compressible or incompressible fluids and low-permeable porous media. Illustrative numerical examples, including random initial data, random boundary conditions or random source terms and a well-known benchmark problem, i.e. Mandel's problem are performed. The results are in good agreement with the theoretical findings. Furthermore, we show numerically that there is a connection between the inf-sup stability of discretizations and the performance of the fixed-stress splitting scheme.