Analysis of Backward Euler Primal DPG Methods

We analyze backward Euler time stepping schemes for a primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in a natural norm and in the L 2 {L^{2}} norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.

A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variableuand the auxiliary variableσwith respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.

FINITE ELEMENT APPROXIMATIONS FOR THE LAPLACE OPERATOR WITH A RIGHT-HAND SIDE MEASURE

We study in this paper existence of a solution for problem Δu=µ in fractional Sobolev spaces [Formula: see text] where Ω is an open bounded polygonal convex domain of ℝ2 and µ a measure on Ω. Thanks to this regularity, we obtain estimates for ∇u−∇uh and u–uh in Lp norm where uh is the elliptic projection on a finite element discretized space associated to the Laplace operator.