Primal dual mixed finite element methods for the elliptic Cauchy problem

TL;DR

This paper introduces primal-dual mixed finite element methods for solving elliptic Cauchy problems, providing error estimates, stability analysis, and a regularization approach that improves convergence and handles data perturbations.

Contribution

It develops a new stabilized primal-dual finite element method with optimal convergence properties and minimal regularization, extending previous least squares approaches.

Findings

Method achieves optimal convergence for smooth solutions.

Regularization does not depend on auxiliary parameters.

Numerical examples confirm theoretical results.

Abstract

We consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Dard\'e, Hannukainen and Hyv\"onen in \emph{An {$H\sb {\sf{div}}$}-based mixed quasi-reversibility method for solving elliptic {C}auchy problems}, SIAM J. Numer. Anal., 51(4) 2013. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.