Gabriel N. Gatica, Cristian Inzunza, Filander A. Sequeira:
New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems
In this paper we propose and analyze Banach spaces-based fully-mixed approaches yielding new finite element methods for numerically solving the coupled partial differential equations describing the pseudostress-assisted diffusion of a solute into an elastic material. Two mixed formulations employing the diffusive flux as an additional variable are introduced for the diffusion equation, and the concentration gradient is considered as an auxiliary unknown of the second one of them. The resulting coupled systems are rewritten as equivalent fixed point operator equations, so that the respective unique solvabilities are proved by applying the classical Banach theorem along with the Babuska-Brezzi theory. The nonlinear dependency on the elastic variables of the diffusion coefficient and its source term, as well as the nonlinear dependency on the concentration of the elastic source term, suggest, for appropriate continuous and discrete analyses, that the unknowns be sought in suitable Lebesgue spaces. The associated Galerkin schemes are addressed similarly, and the Brouwer theorem yields the existence of discrete solutions. A priori error estimates are derived for both approaches, 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 two methods and confirming the theoretical findings, are reported.
This preprint gave rise to the following definitive publication(s):
Gabriel N. GATICA, Cristian INZUNZA, Filander A. SEQUEIRA: New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems. Applied Numerical Mathematics, vol. 193, pp. 148-178, (2023).