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A new primal-dual weak Galerkin (PDWG) finite element method is introducedand analyzed for the ill-posed elliptic Cauchy problems with ultra-lowregularity assumptions on the exact solution. The Euler-Lagrange formulationresulting from the PDWG scheme yields a system of equations involving both theprimal equation and the adjoint (dual) equation. The optimal order errorestimate for the primal variable in a low regularity assumption is established.A series of numerical experiments are illustrated to validate effectiveness ofthe developed theory.

Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Ill-Posed Elliptic Cauchy Problems