A high order discontinuous Galerkin method with Lagrange multipliers for second-order elliptic problems

摘要

A discontinuous Galerkin method with Lagrange multipliers (DGLM) is developed to approximate the solution to the second-order elliptic problems. Lagrange multipliers for the solution and for the flux are considered on the edge/face of each element. The weak gradient and the weak divergence are defined for the elliptic problems. Lagrange multipliers for the solution and for the flux are shown to be the averages of the solutions and the "normal" fluxes at the edge/face, respectively. Unique solvability of the discrete system is proved and an error estimate is derived. The element unknowns are solved in terms of the Lagrange multipliers in element by element fashion. The Schur complement system of the Lagrange multipliers has a block structure, which is kept unchanged while the inside of the blocks gets dense in the higher order approximation. An explanation on algorithmic aspects is given. Some numerical results are presented. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.