We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations ...

In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play important roles in numerical methods for partial differential equations. This article intends to ...

We prove the existence and regularity of the solution of an initial boundary value problem for viscous incompressible non-homogeneous fluids, using a semi-Galerkin approximation and so-called compatibility conditions.

We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u+ ω2u = f on Sn ⊂ Rn+1. Here ∆ is the Laplace-Beltrami operator on Sn, ω is a non-zero constant and f belongs to C2k−2(Sn), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (Sn) and τ > 2k ≥ n/2 + 2 are used to construct an approximate solution. An H1(Sn)error ...

This notes presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed.

Two Galerkin type piecewise polynomial approximation procedures based on bilinear forms with different weight functions are analyzed and compared. Optimal order error estimates are proved and numerical results are presented.

We prove the strong Feller property and ergodicity for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes are forced) via Galerkin approximation approach.

In this paper we analyse the problem of computing eigenvalues and eigenfunctions of the Laplace operator by means of discontinuous Galerkin (DG) methods. It results that several DG methods actually provide a spectrally correct approximation of the Laplace operator. We present here the convergence theory, which applies to a wide class of DG methods, as well as numerical tests demonstrating the t...

The successive Galerkin approximation (SGA) algorithm has recently been developed for approximating solutions to the Hamilton-Jacobi-Isaacs equation. The algorithm produces feedback control laws that are stabilizing on a well-deened region of state space. The objective of this paper is to demonstrate the application of the SGA algorithm to two simple examples. The examples serve several purpose...

In this report, we address several aspects of the approximation of the MHD equations by a Galerkin Discontinuous finite volume schemes. This work has been initiated during a CEMRACS project in July and August 2008 in Luminy. The project was entitled GADMHD (for GAlerkin Discontinuous approximation for the Magneto-Hydro-Dynamics). It has been supported by the INRIA CALVI project. 1 Some properti...