A new primal-dual weak Galerkin finite element method for ill-posed elliptic Cauchy problems

Abstract

A new numerical method is devised and analyzed for a type of ill-posed elliptic Cauchy problems by using the primal–dual weak Galerkin finite element method. This new primal–dual weak Galerkin algorithm is robust and efficient in the sense that the system arising from the scheme is symmetric, well-posed, and is satisfied by the exact solution (if it exists). An error estimate of optimal order is established for the corresponding numerical solutions in a scaled residual norm. In addition, a mathematical convergence is established in a weak L2 topology for the new numerical method. Numerical results are reported to demonstrate the efficiency of the primal–dual weak Galerkin method as well as the accuracy of the numerical approximations.