A local weighted discontinuous Galerkin gradient discretization method for solving ellipticequations is introduced. The local scheme is based on a coarse grid and successively improvesthe solution solving a sequence of local elliptic problems in high gradient regions. Using thegradient discretization framework we prove convergence of the scheme for linear and quasilinearequations under minimal regularity assumptions. The error due to artificial boundary conditionsis also analyzed, shown to be of higher order and shown to depend only locally on the regularityof the solution. Numerical experiments illustrate our theoretical findings and the local method’saccuracy is compared against the non local approach.