New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations

Abstract

In this paper, we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the coupled poroelasticity and heat equations, which describe the interaction between the fields of deformation and temperature. A nonsymmetric pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is an extension of Hooke’s law to account for thermal effects. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each with right-hand terms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška–Brezzi theory in Banach spaces, allows to conclude, under a smallness assumption on the data, the existence of a unique solution. The discrete analysis is conducted in a similar manner, utilizing the Brouwer and Banach theorems to demonstrate both the existence and uniqueness of the discrete solution. The rates of convergence of the resulting Galerkin method are then presented. Finally, a number of numerical tests are shown to validate the aforementioned statement and demonstrate the good performance of the method.