A new constrained formulation of the Maxwell system
A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations
A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations is presented. Applying the orthogonal projection technique, we introduce two local Gauss integrations as a stabilizing term in the error correction method, and derive a new error correction method. In both the coarse solution computation step and the error computation step, a locally stabilizing term based on two local Gauss integrations is introduced. The stability and convergence of the...
A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method
Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of is proved. An -error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations...
A new exponentially fitted triangular finite element method for the continuity equations in the drift-diffusion model of semiconductor devices
In this paper we present a novel exponentially fitted finite element method with triangular elements for the decoupled continuity equations in the drift-diffusion model of semiconductor devices. The continuous problem is first formulated as a variational problem using a weighted inner product. A Bubnov-Galerkin finite element method with a set of piecewise exponential basis functions is then proposed. The method is shown to be stable and can be regarded as an extension to two dimensions of the...
A New family of Mixed Finite Elements in IR3.
A New Family of Stabel Elements for Nearly Incompressible Elasticity Based on a Mixed Petrov-Galerkin Finite Element Formulation.
A new finite element approach for problems containing small geometric details
In this paper a new finite element approach is presented which allows the discretization of PDEs on domains containing small micro-structures with extremely few degrees of freedom. The applications of these so-called Composite Finite Elements are two-fold. They allow the efficient use of multi-grid methods to problems on complicated domains where, otherwise, it is not possible to obtain very coarse discretizations with standard finite elements. Furthermore, they provide a tool for discrete homogenization...
A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method
We consider a coupled PDE model of various fluid-structure interactions seen in nature. It has recently been shown by the authors [Contemp. Math. 440, 2007] that this model admits of an explicit semigroup generator representation 𝓐:D(𝓐)⊂ H → H, where H is the associated space of fluid-structure initial data. However, the argument for the maximality criterion was indirect, and did not provide for an explicit solution Φ ∈ D(𝓐) of the equation (λI-𝓐)Φ =F for given F ∈ H and λ > 0. The present...
A new mixed finite element method based on the Crank-Nicolson scheme for Burgers' equation
In this paper, a new mixed finite element method is used to approximate the solution as well as the flux of the 2D Burgers’ equation. Based on this new formulation, we give the corresponding stable conforming finite element approximation for the pair by using the Crank-Nicolson time-discretization scheme. Optimal error estimates are obtained. Finally, numerical experiments show the efficiency of the new mixed method and justify the theoretical results.
A New Mixed Formulation for Elasticity.
A new quadrilateral MINI-element for Stokes equations
We introduce a new stable MINI-element pair for incompressible Stokes equations on quadrilateral meshes, which uses the smallest number of bubbles for the velocity. The pressure is discretized with the P1-midpoint-edge-continuous elements and each component of the velocity field is done with the standard Q1-conforming elements enriched by one bubble a quadrilateral. The superconvergence in the pressure of the proposed pair is analyzed on uniform rectangular meshes, and tested numerically on uniform...
A new reconstruction-enhanced discontinuous Galerkin method for time-dependent problems
This work is concerned with the introduction of a new numerical scheme based on the discontinuous Galerkin (DG) method. We propose to follow the methodology of higher order finite volume schemes and introduce a reconstruction operator into the DG scheme. This operator constructs higher order piecewise polynomial reconstructions from the lower order DG scheme. Such a procedure was proposed already in [2] based on heuristic arguments, however we provide a rigorous derivation, which justifies the increased...
A Nonconforming Combined Method for Solving Laplace's Boundary Value Problems with Singularities.
A Nonconforming Finite Element Method to Compute the Spectrum of an Operator Relative to the Stability of a Plasma in Toroidal Geometry.
A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem
This study is mainly dedicated to the development and analysis of non-overlapping domain decomposition methods for solving continuous-pressure finite element formulations of the Stokes problem. These methods have the following special features. By keeping the equations and unknowns unchanged at the cross points, that is, points shared by more than two subdomains, one can interpret them as iterative solvers of the actual discrete problem directly issued from the finite element scheme. In this way,...
A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem***
This study is mainly dedicated to the development and analysis of non-overlapping domain decomposition methods for solving continuous-pressure finite element formulations of the Stokes problem. These methods have the following special features. By keeping the equations and unknowns unchanged at the cross points, that is, points shared by more than two subdomains, one can interpret them as iterative solvers of the actual discrete problem directly issued from the finite element scheme. In this way,...
A note on a discrete form of Friedrichs' inequality
The proof of the Friedrichs' inequality on the class of finite dimensional spaces used in the finite element method is given. In particular, the approximate spaces generated by simplicial isoparametric elements are considered.
A note on a dual finite element method
A note on polynomial approximation in Sobolev spaces
A note on polynomial approximation in Sobolev spaces
For domains which are star-shaped w.r.t. at least one point, we give new bounds on the constants in Jackson-inequalities in Sobolev spaces. For convex domains, these bounds do not depend on the eccentricity. For non-convex domains with a re-entrant corner, the bounds are uniform w.r.t. the exterior angle. The main tool is a new projection operator onto the space of polynomials.