In this paper, we consider a two-field nonlinear poroelasticity model, in which the unknown variables are the solid displacement and the pore pressure, and the permeability of the elastic material depends on the dilatation. A standard Galerkin method based on Picard iteration is presented. To reduce the computational costs, we develop the two-grid method, the reduced-order finite element (ROFE) method based on proper orthogonal decomposition (POD) technique, and a combination of the first two methods, i.e., the two-grid reduced-order finite element (TGROFE) method. The associated numerical error has four components due to the coarse triangulation discretization, fine triangulation discretization, time discretization, and eigenvalue truncation by POD. Numerical experiments show the accuracy and efficiency of these methods. To further illustrate the application of the POD technique to the quasi-static poroelasticity problem, we investigate two reduced-order mixed finite element methods, which avoid pressure oscillations and/or have locking-free property.
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