A new family of discontinuous finite element methods for elliptic problems in the whole space

Abstract

We investigate a new hybrid method we call the discontinuous Galerkin‐inverted finite element method (DGIFEM) to approximate the solution of elliptic problems in , especially when the growth or the decay of the solution is very slow. On the basis of both the discontinuous Galerkin discretization and the inverted finite element method, the DGIFEM keeps part of the domain bounded and maps the other infinite extent into a bounded region via a suitable polygonal inversion. The numerical solution is then constructed in an appropriate subspace of weighted Sobolev spaces, where the weights allow the control of the growth or the decay of functions at infinity. A careful study of the convergence of the DGIFEM is carried out and shows that the optimal order of convergence can always be reached. Finally, some numerical results are given as illustration of the good performance of the proposed method.