A gradient weighted moving finite element method (GWMFE) based on piecewise polynomial of any degree is developed to solve time-dependent problems in two space dimensions. Numerical experiments are employed to test the accuracy and effciency of the proposed method with nonlinear Burger equation.

Many problems in science and engineering are formulated in terms of time-dependent partial differential equations (PDEs). It is well known that due to the moving steep fronts present in the solution, these problems present serious numerical difficulties. We present an approach where the mesh moves dynamical to capture the sharp front with a small number of space nodes.

Moving finite-element method (MFE) is a discretization technique on continuously deforming spatial grids introduced by K. Miller and R. N. Miller [

In all of these works, the method is based on a minimization of the PDE residual that is obtained by approximating the solution with piecewise linear elements. In [

Our formulation of GWMFE has been designed to solve a PDE of the type

The GWMFE is a numerical procedure which allows the local gradient adaptation of the finite-element approximation space with time. For the space discretization, we consider a hexagonally connected triangularization of

For example, for node 1 of triangle element with 6 nodes, we have

In GWMFE, this weight function is taken to be

The argument for the use of this weight function is that it de-emphasizes those parts of the integral where

We add the penalization term

The discretization of space-variables transforms each PDE in a system of ODE. To accomplish the discretization of problem (

The approximation

Let us consider a global node

The computation of

Our GWMFE discretization leads to a large ODE system

For the second equation in (

Similarly, the third equation is

Second order terms such as the Laplacian

Based on the idea of smoothing, there are basically three techniques for dealing with this problem. The

So the integral (

Consider the global node

If

However (

This system of ODE can be written as follows:

This system of ODE has a stiff mass matrix and appropriate methods are thus required. In the present work, we use the ODE15S package [

Let time steps of the problem have the form

Now, we apply the refinement scheme at each time step [

In each ODE system, we need the initial conditions which are obtained by solving the previous ODE system. In other words, the initial condition of the

Generally, suppose that we are at time level

The local time step refinement (LTSR) method may be derived as follows

Set

Set

Solve ODE system (

Set

If

If

So our solution,

We present a numerical example to illustrate the performance of our GWMFE. The integrals that appear in the system of ODE, say (

Some of the more difficult and interesting real life problems in which adaptive algorithms are needed arise in transport phenomena in which steep fronts propagate through the domain. The special case of the nonlinear Burger equation is often used to test numerical methods so we consider the nonlinear evolution equation

Figure

Nodes movement for Burger equation from

Nodes distribution and solution of Burgers equation with quadratic elements for uniform initial mesh.

Nodes movement and related solution of Burgers equation with quadratic elements at

Nodes movement and related solution of Burgers equation with quadratic elements at

Nodes movement and related solution of Burgers equation with quadratic elements at

Let us consider to the average error,

Table

CPU time, number of function evaluation (NFE), and average error (

Number of mesh nodes | CPU time | NFE | |
---|---|---|---|

2423.453600 | 384 | 0.62632852 | |

7687.359375 | 746 | 0.44484591 | |

13839.750000 | 1039 | 0.34352671 |

In this paper, we presented a gradient weighted moving finite-element method based on polynomial approximations of high degree for the solution of time-dependent PDEs on two-dimensional space domains. We used a solution-dependent weight function for original MFE formulation to have better performance and mesh adaptivity. These moving nodes method are especially suited to problems which develop sharp moving fronts, especially problems where one needs to resolve the fine-scale structure of the fronts.

A careful treatment of the general second order terms is carried out. Moreover, by using numerical evaluations of all integrals, we can solve a large class of problems without extra calculations. The GWMFE is applied to the Burger test equation for transport process with quadratic polynomial as interpolation function. One can solve this problem with other nonlinear approximation function as well as other penalty constants. Numerical results are given to illustrate the good behavior of the GWMFE when using some cases of penalty constants.