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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.
A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as...
A higher order pressure segregation scheme for the time-dependent incompressible magnetohydrodynamics (MHD) equations is presented. This scheme allows us to decouple the MHD system into two sub-problems at each time step. First, a coupled linear elliptic system is solved for the velocity and the magnetic field. And then, a Poisson-Neumann problem is treated for the pressure. The stability is analyzed and the error analysis is accomplished by interpreting this segregated scheme as a higher order...
We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the...
An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived...
In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H) | where H is the coarse mesh size. Without any assumptions...
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