Primal-Dual Weak Galerkin Finite Element Methods for Elliptic Cauchy Problems
This work provides a novel numerical method for solving ill-posed elliptic Cauchy problems, which are challenging in computational mathematics.
The authors propose a well-posed numerical scheme for ill-posed elliptic Cauchy problems using a constrained minimization approach with weak Galerkin finite element methods, achieving symmetric and consistent primal-dual formulations with stability and error estimates.
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.