A discontinuous Petrov--Galerkin method with Lagrangian multipliers for second order elliptic problems

We present a discontinuous Petrov–Galerkin (DPG) method for the finite element discretization of second order elliptic boundary value problems. The novel approach emanates from a one-element weak formulation of the differential problem. This procedure, which is typical of discontinuous Galerkin (DG) methods, is based on introducing variables defined in the interior and on the boundary of the element. The interface variables are suitable Lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative, thus providing the proper connection between neighboring elements. The internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size in the sole Lagrangian multipliers. A stability and convergence analysis of the novel formulation is carried out and its connection with mixed-hybrid and DG methods is explored. Numerical tests on several benchmark problems are included to validate the convergence performance and the flux-conservation properties of the DPG method.