A Pseudostress-Based Mixed-Primal Finite Element Method for Stress-Assisted Diffusion Problems in Banach Spaces

Abstract

In this paper we consider the system of partial differential equations describing the stress-assisted diffusion of a solute into an elastic material, and introduce and analyze a Banach spaces-based variational approach yielding a new mixed-primal finite element method for its numerical solution. The elasticity model involved, which is initially defined according to the constitutive relation given by Hooke’s law, and whose momentum equation holds with a concentration-depending source term, is reformulated by using the non-symmetric pseudostress tensor and the displacement as the only unknowns of the associated mixed scheme, in addition to assuming a Dirichlet boundary condition for the latter. In turn, the diffusion equation, whose diffusivity function and source term depend on the stress and the displacement of the solid, respectively, is set in primal form in terms of the concentration unknown and a Dirichlet boundary condition for it as well. The resulting coupled formulation is rewritten as an equivalent fixed point operator equation, so that its unique solvability is established by employing the classical Banach theorem along with the corresponding Babuška-Brezzi theory and the Lax-Milgram theorem. The aforementioned dependence of the diffusion coefficient and the subsequent treatment of this term in the continuous analysis, suggest to better look for the solid unknowns in suitable Lebesgue spaces. The discrete analysis is performed similarly, and the Brouwer theorem yields existence of a Galerkin solution. A priori error estimates are derived, and rates of convergence for specific finite element subspaces satisfying the required discrete inf-sup conditions, are established in 2D. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical convergence, are reported.