Stable discontinuous Galerkin FEM without penalty parameters

We propose a modified local discontinuous Galerkin (LDG) method for second--order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincar\'e--Friedrichs inequality for the discrete gradient that employs a lifting of the jumps with one polynomial degree higher than the scalar approximation space. Our analysis covers rather general simplicial meshes with the possibility of hanging nodes.Comment: Accepted for publication in the conference proceedings of Numerical Mathematics and Advanced Applications ENUMATH 2015. Typo correcte

Discontinuous Galerkin Methods for Second-Order Elliptic PDE with Low-Regularity Solutions

In this paper we derive an a priori error analysis for interior penalty discontinuous Galerkin finite element discretizations of the Poisson equation with exact solution inW 2,p , p∈(1,2]. We show that the DGFEM converges at an optimal algebraic rate with respect to the number of degrees of freedo

Substructuring Preconditioners for h-p Mortar FEM

International audienceWe build and analyze a substructuring preconditioner for the mortar method in the h-p finite element framework. Particular attention is given to the construction of the coarse component of the preconditioner in this framework, in which continuity at the cross points is not required. Two variants are proposed: the first one is an improved version of a coarse preconditioner already presented in [12]. The second is new and is built by using a Discontinuous Galerkin interior penalty method as coarse problem. A bound of the condition number is proven for both variants and their efficiency and scalability is illustrated by numerical experiments

Nitsche's method for a Robin boundary value problem in a smooth domain

We prove several optimal-order error estimates for the finite element method applied to an inhomogeneous Robin boundary value problem for the Poisson equation defined in a smooth bounded domain in $\mathbb{R}^n$, $n=2,3$. The boundary condition is imposed weakly by the Nische's method. We also study the symmetric interior penalty discontinuous Galerkin method and prove the same error estimates. Numerical examples to confirmed our results are also reported.Comment: 16 pages, 7 figure