Analysis of the element-free Galerkin method with penalty for general second-order elliptic problems

Abstract In this paper, an element-free Galerkin (EFG) method with penalty for solving general second-order elliptic problems with mixed boundary conditions is presented. A priori approximate estimations of the penalty method are obtained under some proper assumptions, which ensure the availability of the penalty method. By defining a special norm, the existence and uniqueness for the weak solution of the penalized continuous variational formula are proved, which guarantee the validity of the EFG discretization. Error estimates of the EFG method are provided in L2 and H1 norms for the Dirichlet and the mixed boundaries, respectively. Numerical examples are finally performed to illustrate the theoretical results.