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

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.

27 pages, 4 figures

References in corpus (2)

• On level set type methods for elliptic Cauchy problems
• A Primal-Dual Weak Galerkin Finite Element Method for Fokker-Planck Type Equations

Cited by in corpus (3)

• A primal-dual finite element method for first-order transport problems
• The Basics of Weak Galerkin Finite Element Methods
• A New Primal-Dual Weak Galerkin Finite Element Method for Ill-posed Elliptic Cauchy Problems