We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler scheme in time. Our estimates are based on a H-conforming reconstruction of the potential, continuous and piec...

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the time discontinuous Galerkin solution of linear parabolic equations. Such estimates have many applications. They are essential, for example, in establishing optimal a priori error estimates in nonHilbertian norms without unnatural coupling of spatial mesh sizes with tim...

A mixed finite element method for the Navier–Stokes equations is introduced in which the stress is a primary variable. The variational formulation retains the mathematical structure of the Navier–Stokes equations and the classical theory extends naturally to this setting. Finite element spaces satisfying the associated inf–sup conditions are developed. Mathematics Subject Classification. 65N60,...

Abstract. In this paper we introduce and analyze two augmented mixed finite element methods for a velocity-pressure-stress formulation of the stationary Stokes equations. Our approach, which extends analogue results for linear elasticity problems, is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boun...

In this work, we present a cell-centered time-splitting technique for solving evolutionary diffusion equations on triangular grids. To this end, we consider three variables (namely the pressure, the flux and a weighted gradient) and construct a so-called expanded mixed finite element method. This method introduces a suitable quadrature rule which permits to eliminate both fluxes and gradients, ...

In this article, we develop and analyze a mixed finite element method for the Stokes equations. Our mixed method is based on the pseudostress-velocity formulation. The pseudostress is approximated by the RaviartThomas (RT) element of index k ≥ 0 and the velocity by piecewise discontinuous polynomials of degree k. It is shown that this pair of finite elements is stable and yields quasi-optimal a...

A physically based duality theory for first order elliptic systems is shorn to be of central importance in connection with the Galerkin finite element solution of these systems. Using this theory in conjunction with a certain hypothesis concerning approximation spaces, optimal error estimates for Galerkin type approximations are demonstrated. An example of a grid which satisfies the hypothesis ...

A new mixed nite element method on totally distorted rectangular meshes is introduced with optimal error estimates for both pressure and velocity This new mixed discretization ts the geometric shapes of the discontinuity of the rough coe cients and domain boundaries well This new mixed method also enables us to derive the optimal error estimates and existence and uniqueness of Thomas s mixed ni...

Small deformations of a viscoelastic body are considered through the linear Maxwell and Kelvin-Voigt models in the quasi-static equilibrium. A robust mixed finite element method, enforcing the symmetry of the stress tensor weakly, is proposed for these equations on simplicial tessellations in two and three dimensions. A priori error estimates are derived and numerical experiments presented. The...

Various applications in fluid dynamics and computational continuum mechanics motivate the development of reliable and efficient adaptive algorithms for mixed finite element methods. In order to save degrees of freedom, not all but just a selection of finite element domains are refined. Hence the fundamental question of convergence as well as the question of optimality require new mathematical a...