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 L2 topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.