In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show tha...

We study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the H(div)-norm. This new element is designed so that the H(div)-projection Ph satisfiesr Ph = Phdiv. A rigorous optimal order error estimate is carried out by proving a modified version of the Bramble–Hilbert lemma for vector variables. We show that a local H(div)-projection r...

The extended framework of Hamilton's principle and the mixed convolved action principle provide new rigorous weak variational formalism for a broad range of initial boundary value problems in mathematical physics and mechanics. In this paper, their potential when adopting temporally higher order approximations is investigated. The classical single-degree-of-freedom dynamical systems are primari...

We consider mixed nite element methods for second order elliptic equations on non-matching multiblock grids. A mortar nite element space is introduced on the non-matching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of ux condition. A standard mixed nite element method is used within the blocks. Optimal order convergence is shown f...

In this dissertation, we consider new approaches to the construction of meshes, discretization, and preconditioning of the resulting algebraic systems for the diffusion equation with discontinuous coefficients. In the first part, we discuss mixed finite element approximations of the diffusion equation on general polyhedral meshes. We introduce a non-conforming approximation method for the flux ...

The numerical solution of Dirichlet's problem for a second order elliptic operator in divergence form with arbitrary nonlinearities in the rst and zero order terms is considered. The mixed nite element method is used. Existence and uniqueness of the approximation are proved and optimal error estimates in L are demonstrated for the relevant functions. Error estimates are also derived in L, 2 q +...

The progress made during the past decade in the application of mixed finite element methods to solve viscoelastic flow problems using differential constitutive equations is reviewed. The algorithmic developments are discussed in detail. Starting with the classical mixed formulation, the elastic viscous stress splitting (EVSS) method as well as the related discrete EVSS and the so-called EVSS-G ...

For least-squares mixed nite element methods for the rst-order system formulation of second-order elliptic problems, a technique for the weak enforcement of boundary conditions is presented. This approach is based on least-squares boundary functionals which are equivalent to the H ?1=2 and H 1=2 norms on the trace spaces of lowest-order Raviart-Thomas elements for the ux and standard continuous...

A new mixed variational formulation of the equations of stationary incompressible magneto–hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard ...

In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity a...