Finite Element Approximation of a Hemivariational Inequality for Steady-State Heat Conduction: Double-Limit Convergence of Penalization and Discretization

Abstract: In this paper we study the numerical approximation and convergence analysis of a steady-state heat conduction problem with mixed boundary conditions. The physical model is governed by a hemivariational inequality depending on a heat transfer parameter $α> 0$. We consider the finite element approximation of this penalized problem, as well as its corresponding limit problem with a prescribed constant temperature on a part of the boundary. The main theoretical result establishes the strong convergence of the discrete solutions. Specifically, we prove the double-limit convergence of the finite element approximations to the limit solution as the mesh size $h$ tends to zero and the penalization parameter $α$ tends to infinity, independently and simultaneously. We further complement the convergence analysis with an error estimate of optimal order for the discrete limit problem, and we explain why an estimate uniform in the penalization parameter cannot be expected. The theoretical results are illustrated by numerical simulations on four examples verified by the method of manufactured solutions.