Convergence of an Adaptive Finite Element Method for Distributed Flux Reconstruction

TL;DR

This paper proves the convergence of an adaptive finite element method for reconstructing distributed fluxes in diffusion systems, ensuring the method's reliability and accuracy through a posteriori error estimators.

Contribution

It establishes the convergence of an adaptive conforming finite element method for distributed flux reconstruction, based on a posteriori error estimators, which was previously unproven.

Findings

Sequence of solutions converges to the true solution in energy norm.

Error estimators asymptotically approach zero.

Method guarantees reliable flux reconstruction.

Abstract

We shall establish the convergence of an adaptive conforming finite element method for the reconstruction of the distributed flux in a diffusion system. The adaptive method is based on a posteriori error estimators for the distributed flux, state and costate variables. The sequence of discrete solutions produced by the adaptive algorithm is proved to converge to the true triplet satisfying the optimality conditions in the energy norm and the corresponding error estimator converges to zero asymptotically.