Raytcho D. Lazarov

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems

We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continuous Galerkin, nonconforming, and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric, and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain, which are then automatically coupled. Finally, the framework brings about a new point of view, thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.

First-order system least squares for second-order partial differential equations: part I

This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in

A least-squares approach based on a discrete minus one inner product for first order systems

n=2

Least-squares mixed finite elements for second-order elliptic problems

or

Finite volume methods for convection-diffusion problems

3

Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations

dimensions as a system of first-order equations. In part I [Z. Cai, R. D. Lazarov, T. Manteuffel, and S. McCormick, SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799] a similar functional was developed and shown to be elliptic in the

Superconvergence of the velocity along the Gauss lines in mixed finite element methods

H(\divv) \times H^1

Finite volume element approximations of nonlocal reactive flows in porous media

norm and to yield optimal convergence for finite element subspaces of

The Galerkin finite element method for a multi-term time-fractional diffusion equation

H(\divv) \times H^1

Variational formulation of problems involving fractional order differential operators

. In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the