A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Convection--Diffusion

TL;DR

This paper presents a novel least-squares weak Galerkin finite element method for solving severely ill-posed convection--diffusion Cauchy problems, offering stability, flexibility, and optimal error estimates.

Contribution

It introduces a new LS-WG scheme that transforms the problem into a symmetric positive definite system and proves its convergence and robustness.

Findings

The method yields optimal-order error estimates.

Numerical tests confirm theoretical convergence rates.

The scheme is robust and efficient on complex geometries.

Abstract

We introduce and rigorously analyze a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem of convection--diffusion equations. The proposed framework utilizes weak derivatives defined on a class of discontinuous weak functions, enabling the natural treatment of complex boundary conditions and internal interfaces. A key advantage of the least-squares formulation is that it transforms the underlying non-self-adjoint operator into a discrete linear system that is inherently symmetric and positive definite (SPD). We demonstrate the geometric flexibility of the method on arbitrary polygonal and polyhedral partitions. Furthermore, we establish the uniqueness of the numerical solution and derive optimal-order error estimates in a carefully defined discrete energy norm. Extensive numerical tests are presented to confirm the theoretical convergence rates and highlight the algorithm's robustness and efficiency compared to standard Galerkin approaches.