Primal-Dual Weak Galerkin Finite Element Methods for Elliptic Cauchy Problems

TL;DR

This paper introduces a novel primal-dual weak Galerkin finite element method for solving ill-posed elliptic Cauchy problems, ensuring well-posedness, stability, and providing error estimates with numerical validation.

Contribution

It develops a new primal-dual weak Galerkin scheme that is symmetric, well-posed, and consistent for elliptic Cauchy problems, with theoretical analysis and numerical validation.

Findings

The method is symmetric and well-posed.

Error estimates are established in discrete Sobolev norms.

Numerical results confirm the theoretical analysis.

Abstract

The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler-Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal-dual weak Galerkin finite element method. This new primal-dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak $L^2$ topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.