Continuous Galerkin Method Research Articles

Calculation of Transient Temperature Fields with Finite Elements in Unstructured Space and Time Dimensions.

The stability of time-stepping methods for parabolic differential equations is critical issue. Furthermore, solving such equations with a classical time-stepping approach can be very expensive because many small time steps have to be taken if steep gradients occur in the solution, even if they occur only in a small part of the space domain. In this paper we present a discretization technique in which finite element approximations are used in time and space simultaneously for a relatively large time period called a time slab. The weighted residual process is used to formulate a finite element method for a space-time domain based upon the continuous Galerkin method. This technique may be repeatedly applied to obtain further parts of the solution in subsequent time intervals.

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N-1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.

Finite element simulations for compressible miscible displacement with molecular dispersion in porous media

We consider a nonlinear parabolic system describing compressible miscible displacement in a porous medium [5]. Continuous time and discrete time Galerkin methods are introduced to approximate the solution and optimalH1 error estimates are obtained. One contribution of this paper is a demonstration of how molecular dispersion can be handled.

Mixed methods for compressible miscible displacement with the effect of molecular dispersion

We consider a nonlinear parabolic system describing compressible miscible displacement in a porous medium[4]. Continuous time and discrete time Galerkin methods are introduced to approximate the solution, and optimalH 1-error estimates are obtained. This paper improves upon previously derived estimates in two aspects. Firstly, error estimates are given with no restrictions on the diffusion tensor. That is, we have included the effects of molecular diffusion and dispersion. Secondly, the complete compressible case is considered in the error analysis.

Behaviour in the large of numerical solutions to one-dimensional nonlinear viscoelasticity by continuous time Galerkin methods

We analyze the long time behaviour of fully discrete solutions to a one-dimensional nonlinear viscoelastic problem. It is shown that these approximations which are found by a continuous time Galerkin method converge to a steady state. The possible numerical steady states are characterized and in particular their high degree of dependence on initial data and mesh design is explained. Computational results are included which show the above dependence and indicate that the numerical solutions will typically not converge to unstable states.

Condition Estimates

Condition Estimates

On the Numerical Solution of Parabolic Equations in a Single Space Variable by the Continuous Time Galerkin Method

We consider the Galerkin method to solve a parabolic initial boundary value problem in one space variable, using piecewise polynomial functions and give an alternative proof of superconvergence. Then by means of Lobatto quadrature, we obtain purely explicit vector initial value problems without loss in the order of accuracy, global or pointwise.