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A new finite element, which is continuously differentiable,
but only piecewise quadratic
polynomials on a type of uniform triangulations, is introduced.
We construct a local basis which
does not involve nodal values nor derivatives.
Different from the traditional finite elements, we have to
construct a special, averaging operator
which is stable and preserves quadratic polynomials.
We show the optimal order of approximation
of the finite element in interpolation, and in solving
the biharmonic...
A continuous finite element method to approximate Friedrichs' systems is
proposed and analyzed. Stability is achieved by penalizing the jumps
across mesh
interfaces of the normal derivative of some components of the discrete solution.
The convergence analysis leads to optimal convergence rates
in the graph norm and suboptimal of order ½ convergence rates in
the L2-norm. A variant of the method specialized to
Friedrichs' systems associated with elliptic PDE's in mixed form and
reducing the number...
For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present...
We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for
nearly and perfectly incompressible linear elasticity. These mixed methods allow the
choice of polynomials of any order k ≥ 1 for the approximation of the
displacement field, and of order k or k − 1 for the
pressure space, and are stable for any positive value of the stabilization parameter. We
prove the optimal convergence of the displacement and stress fields...
We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k ≥ 1 for the approximation of the displacement field, and of order k or k − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields in both cases, with error estimates that are independent...
We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for
nearly and perfectly incompressible linear elasticity. These mixed methods allow the
choice of polynomials of any order k ≥ 1 for the approximation of the
displacement field, and of order k or k − 1 for the
pressure space, and are stable for any positive value of the stabilization parameter. We
prove the optimal convergence of the displacement and stress fields...
A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference...
From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.
From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.
In this paper we propose a finite element method for the approximation of second order elliptic problems on composite grids. The method is based on continuous piecewise polynomial approximation on each grid and weak enforcement of the proper continuity at an artificial interface defined by edges (or faces) of one the grids. We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions, and present some numerical examples.
In this paper we propose a finite element method for the approximation of
second order elliptic problems on composite grids. The method is
based on continuous piecewise polynomial approximation on each
grid and weak enforcement of the proper continuity at an
artificial interface defined by edges (or faces) of one the grids.
We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions,
and present some numerical examples.
We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume...
We present a finite volume method based on the integration of the Laplace
equation on both the cells of a primal almost arbitrary two-dimensional
mesh and those of a
dual mesh obtained by joining the centers of the cells of the primal mesh.
The key ingredient is the definition of discrete gradient and divergence
operators verifying a discrete Green formula.
This method generalizes an existing finite volume method that
requires “Voronoi-type” meshes.
We show the equivalence of this finite volume...
A standard method for proving the inf-sup condition implying stability of
finite element approximations for the stationary Stokes equations is to
construct a Fortin operator. In this paper, we show how this can be done
for two-dimensional triangular and rectangular Taylor-Hood methods, which
use continuous piecewise polynomial approximations for both velocity and
pressure.
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