# A block preconditioner for a mixed finite element method for Biot's equations

#### Abstract

In this thesis, we explore the solution methods for the linear system resulting from a mixed finite element method applied to the Biot's consolidation model. This model describes the coupled interactions between a porous solid and the fluid contained within it. Specifically, we use a method developed by Yi [Numer. Methods for PDEs, 29(5), pp. 1749-1777] that expands Biot's system to include fluid pressure, solid displacement, fluid flux and total stress as primary unknowns. As the resulting linear system is a large, sparse, saddle point system, we attempt to solve this system via a Schur complement preconditioned iterative method. Using the exact Schur complement preconditioner would require the inversion of the first block of the saddle point system A. Since this can still be computationally expensive, we attempt to use an approximation to the Schur complement based on a spectrally equivalent approximation to A. To test the preconditioner, we solve problems in homogeneous and heterogeneous layered media. In the homogeneous case, we show that the number of iterations required to solve the system increases only slightly when the element size and time step are decreased at corresponding rates. In the case of heterogeneous material, we require slightly more iterations to solve the problem as the difference in the material parameters of layers is more pronounced. However, the amount of work required to apply the preconditioner within each iteration seems to depend on the difference in the material parameters of the layers and the size of the element.

#### Subject Area

Applied Mathematics

#### Recommended Citation

Bean, Maranda, "A block preconditioner for a mixed finite element method for Biot's equations" (2014). *ETD Collection for University of Texas, El Paso*. AAI1583895.

https://scholarworks.utep.edu/dissertations/AAI1583895