Enriched finite element method for Poisson equations

We develop an enriched finite element method for the Poisson equation with Dirichlet boundary conditions. Starting from the standard linear Lagrangian finite element, the local approximation space is enriched by three functions associated with edge-average degrees of freedom. The construction is presented within the Ciarlet framework, and the conditions ensuring local admissibility and global conformity are examined. Two new two-parameter families of weighted enrichment functions, based on Jacobi-type weights, are introduced, and explicit expressions for the corresponding local basis functions are derived. For one of these families, an explicit L2-error estimate is obtained for the enriched interpolation operator, with a constant depending on the enrichment parameters. For conforming enriched spaces containing the standard linear finite element space, the error of the enriched Galerkin approximation in the energy norm is proved to be no greater than that of the standard linear approximation. The numerical study compares the proposed enriched elements with the standard linear and quadratic Lagrangian finite elements on regular and nonuniform triangulations. The numerical results show that the most effective enrichment, as well as the choice of the Jacobi weight parameters, depends on the problem under consideration. The numerical examples include heat-conduction problems and a prescribed-crack problem with a singular solution at the crack tip.