A high-order discontinuous Galerkin method with Lagrange multipliers for advection–diffusion problems

A high-order Discontinuous Galerkin method with Lagrange Multipliers (DGLM) is presented for the solution of advection–diffusion problems on unstructured adaptive meshes. Unlike other hybrid discretization methods for transport problems, it operates directly on the second-order form the advection–diffusion equation. Like the Discontinuous Enrichment Method (DEM), it chooses the basis functions among the free-space solutions of the homogeneous form of the governing partial differential equation, and relies on Lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries. However unlike DEM, the proposed hybrid discontinuous Galerkin method approximates the Lagrange multipliers in a space of polynomials, instead of a space of traces on the element boundaries of the normal derivatives of a subset of the basis functions. For a homogeneous problem, the design of arbitrarily high-order elements based on this DGLM method is supported by a detailed mathematical analysis. For a non-homogeneous one, the approximated solution is locally decomposed into its homogeneous and particular parts. The homogeneous part is captured by the DGLM elements designed for a homogenous problem. The particular part is obtained analytically after the source term is projected onto an appropriate polynomial space. This decoupling between the two parts of the solution is another differentiator between DGLM and DEM with attractive computational advantages. An a posteriori error estimator for the proposed method is also derived to enable adaptive mesh refinement. All theoretical results are illustrated by numerical simulations that furthermore highlight the potential of the proposed high-order hybrid DG method for transport problems with steep gradients in the high Péclet number regime.