A Finite Element Method for Elliptic Hemivariational Inequalities in Non-isotropic and Heterogeneous Semipermeable Media

TL;DR

This paper develops finite element methods for complex elliptic hemivariational inequalities in non-isotropic, heterogeneous semipermeable media, providing theoretical analysis and numerical validation.

Contribution

It extends existing models to include non-isotropic and heterogeneous media, establishing solution existence, uniqueness, and optimal error estimates.

Findings

Proved existence and uniqueness of solutions.

Derived optimal a priori error estimates.

Numerical experiments confirmed theoretical convergence rates.

Abstract

This work investigates finite element approximations for a general class of elliptic hemivariational inequalities arising in semipermeable media. The proposed model incorporates non-isotropic and heterogeneous diffusion coefficients, alongside both interior and boundary semipermeability terms, extending the isotropic and homogeneous framework examined by Han (2019). The existence and uniqueness of solutions are rigorously established. An optimal a priori error estimate for the linear finite element approximation is derived under appropriate solution regularity assumptions. Numerical experiments are presented to corroborate the theoretical analysis and to confirm the optimal convergence rates for the non-isotropic and heterogeneous case.