Order Galerkin Method Research Articles

Effects of a Magnetic Modulation on the Stability of a Magnetic Liquid Layer Heated From Above

The effect of a time-sinusoidal magnetic field on the onset of convection in a horizontal magnetic fluid layer heated from above and bounded by isothermal non magnetic boundaries is investigated. The analysis is restricted to static and linear laws of magnetization. A first order Galerkin method is performed to reduce the governing linear system to the Mathieu equation with damping term. Therefore, the Floquet theory is used to determine the convective threshold for the free-free and rigid-rigid cases. With an appropriate choice of the ratio of the magnetic and gravitational forces, we show the possibility to produce a competition between the harmonic and subharmonic modes at the onset of convection.

The Generalized Nonlinear Galerkin Method Applied to Hydrodynamics of the Open Sea

In this paper we present new numerical algorithms based on a generalized nonlinear Galerkin method in order to solve coastal and oceanic circulation problems. The equations system is based on the primitive equations of the ocean under Boussinesq and hydrostatic approximations. These equations are transformed using, at the same time, the classical σ transformation and an original homogenization of the boundary conditions. We use a well adapted special basis to apply the usual Galerkin method and the nonlinear Galerkin method. This basis is built on a modelization of the energetic transfers through the different scales of flow. Two approaches are proposed to solve the continuity equation: the (nonlinear) Galerkin method and the method of the characteristics. We present the advantages and drawbacks of both methods.

A novel implementation of multilevel fast multipole algorithm for higher order Galerkin's method

A new approach is proposed to reduce the memory requirements of the multilevel fast multipole algorithm (MLFMA) when applied to the higher order Galerkin's method. This approach represents higher order basis functions by a set of point sources such that a matrix-vector multiply is equivalent to calculating the fields at a number of points from given current sources at these points. The MLFMA is then applied to calculate the point-to-point interactions. This permits the use of more levels in MLFMA than applying MLFMA to basis-to-basis interactions directly and, thus, reduces the memory requirements significantly.

An optimal three-field finite element approximation of the Stokes system with continuous extra stresses

The aim of this paper is to present a mixed finite element method for solving the twodimensional Stokes system, written in terms of the variables velocity, pressure and extra stress tensor. A convenient functional framework is introduced, allowing simpler choices of the type of approximation for the third variable. This is assumed here to be continuous, in order to ensure its applicability to viscoelastic flow problems. In the particular second order Galerkin method that is studied, the dimension of the corresponding finite element subspace is reduced to a minimum, at least as far as local stability analyses are concerned.

A Galerkin method using rational basis functions

Rational basis functions were constructed without any poles. These basis functions were applied to the Galerkin method in order to solve a wide class of differential equations.

Upwinding of high order Galerkin methods in conduction−convection problems

AbstractUpwinded parabolic and cubic elements are derived on a uniform grid of size h for the finite element Galerkin method applied to the solution of the model conduction−convection problem εu″ — Ku′ = 0, ε, K > 0, subject to the boundary conditions u(0) = 1, u (1) = 0. Extension of the results to more complicated problems is indicated. Finally numerical results are given comparing some of the methods derived for a range of L(=Kh/2ε), the grid Peclet number.

High order local approximations to derivatives in the finite element method

Consider the approximation of the solution u of an elliptic boundary value problem by means of a finite element Galerkin method of order r, so that the approximate solution u h {u_h} satisfies u h − u = O ( h r ) {u_h} - u = O({h^r}) . Bramble and Schatz (Math. Comp., v. 31, 1977, pp. 94-111) have constructed, for elements satisfying certain uniformity conditions, a simple function K h {K_h} such that K h ∗ u h − u = O ( h 2 r − 2 ) {K_h}\; \ast \;{u_h} - u = O({h^{2r - 2}}) in the interior. Their result is generalized here to obtain similar superconvergent order interior approximations also for derivatives of u.