A Galerkin Method for Biot Consolidation Model

Abstract

The main aim of this paper is to prove the existence and uniqueness of solutions to an initial-boundary value problem corresponding to the Biot model. The existence theorem is proved by Galerkin method and the passage to the limit in the approximation process is shown in a standard way. Assuming that the given data satisfy some natural regularity requirements a better regularity of solutions is obtained than it could be found in the literature.

Analysis of Nonlinear Poro-Elastic and Poro-Visco-Elastic Models

“…Additionally, uniqueness comes easily in the linear dynamics and does not depend on the regularity properties of the solution nor the dimension of the space; this is certainly not the case for the dynamics considered here. A linear elastic version of the model in Section 2 is also considered in [40], but with a different (and stronger) notion of solution than that in [55]. In [40] a Galerkin method is proposed for purely homogeneous boundary conditions for the pressure and displacement.…”

“…A linear elastic version of the model in Section 2 is also considered in [40], but with a different (and stronger) notion of solution than that in [55]. In [40] a Galerkin method is proposed for purely homogeneous boundary conditions for the pressure and displacement. This allows for a nice notion of strong solution coming from the viability of smoother test functions.…”

“…We topologize the space V via a(·, ·), which is to say that we take the norm induced by a(·, ·) as the norm on V (see Assumption 3.1 below). In the case of constant permeability, the model is linear, as in [40], and one has access to both a "strong" notion of solution (classically differentiable in time) and a "weak" notion of solutions (only distributionally differentiable in time). Here, our notion of solution depends critically on the parameter δ .…”

“…When δ > 0, it is needed to identify the weak limit of the sequence in (50). In contrast, when δ = 0, it is used to perform the summation by parts in the time-discretized pressure equation (40). (25)- (26).…”

“…In [40], a method based on the Galerkin approximation is utilized. However, the physical boundary conditions we consider prevent us from easily solving the ODE system associated with (15)- (22), despite the fact that, at least for δ > 0, we obtain sufficient regularity for u t (i.e., u is differentiable in time into (L 2 (Ω )) 3 ).…”