Isabelle Greff

The AL Basis for the Solution of Elliptic Problems in Heterogeneous Media

In this paper, we will show that, for elliptic problems in heterogeneous media, there exists a local (generalized) finite element basis (AL basis) consisting of

Lagrangian for the convection–diffusion equation

O\big( \big( \log\frac{1}{H}\big) ^{d+1}\big)

Nonconforming Box-Schemes for Elliptic Problems on Rectangular Grids

basis functions per nodal point such that the convergence rates of the classical finite element method for Poisson-type problems are preserved. Here H denotes the mesh width of the finite element mesh and d is the spatial dimension. We provide several numerical examples beyond our theory, where even

Numerical solution of the Poisson equation on domains with a thin layer of random thickness

O(1)

On least action principles for discrete quantum scales

basis functions per nodal point are sufficient to preserve the convergence rates.

Numerical Method for Elliptic Multiscale Problems

Using the asymmetric fractional calculus of variations, we derive a fractional Lagrangian variational formulation of the convection–diffusion equation in the special case of constant coefficients. Copyright

Numerical solution of the homogeneous Neumann boundary value problem on domains with a thin layer of random thickness

Recently, Courbet and Croisille [RAIRO Mode´l. Math. Anal. Nume´r., 32 (1998), pp. 631-649] introduced the finite volume box-scheme for the two-dimensional (2D) Poisson problem in the case of triangular meshes. Generalizations to higher degree box-schemes have been published by Croisille and Greff [Numer. Methods Partial Differential Equations, 18 (2002), pp. 355-373]. These box-schemes are based on the principle of the finite volume method in that they take the average of the equations on each cell of the grid. This gives rise to a natural choice of unknowns located at the interface of the mesh. These box-schemes are conservative and use only one mesh. They can be interpreted as a discrete mixed Petrov-Galerkin formulation of the Poisson problem. In this paper we focus our interest on box-schemes for the Poisson problem in two dimensions on rectangular grids. We discuss the basic finite volume box-scheme and analyze and interpret it as three different box-schemes. The method is demonstrated by numerical examples.

Discrete Embeddings for Lagrangian and Hamiltonian Systems

The present article is dedicated to the numerical solution of the Poisson equation on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on a random domain is transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Robin boundary condition which yields a third order accurate solution in the scale parameter

A continuous/discrete fractional Noether's theorem

\varepsilon

Some nonconforming mixed box schemes for elliptic problems

of the layers thickness. With the help of the Karhunen--Loeve expansion, we transform this random boundary value problem into a deterministic parametric one with a possibly high-dimensional parameter