A New Primal-Dual Weak Galerkin Finite Element Method for Ill-posed Elliptic Cauchy Problems
Provides a robust numerical method for solving ill-posed elliptic Cauchy problems, which are challenging in computational mathematics.
The paper introduces a primal-dual weak Galerkin finite element method for ill-posed elliptic Cauchy problems, achieving a symmetric and well-posed system with optimal-order error estimates in a scaled residual norm and weak L^2 convergence. Numerical results confirm the method's efficiency and accuracy.
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 $L^2$ 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.