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Comparison with the classical FEM

To prove the advantage of the X-FEM, we have computed some results to compare with the classical finite element method.

The stationary mesh:
The stationary mesh method consists of a classical finite element approximation with a fixed mesh over time. For the case here of one-dimension, the mesh is composed by linearly equally spaced points. The grid does not depend on the position of the front during time. Each element is a segment, its length is where represents the length of the domain and the number of elements. This method does not need to rebuild the mass and stiffness matrices every time step.

Figure 9: The unmoving mesh.

The moving mesh:

The moving mesh method consists of remeshing the domain at each time step depending upon the position of the front during time. The mesh is composed by a coarse background mesh and a fine mesh in the vicinity of the front. In our case, the coarse mesh is composed by 20 elements, the region where the mesh is thiner is .

Figure 10: The moving mesh at an initial time step.
Figure 11: The moving mesh at a subsequent time step.

Results: the comparison between the methods:

The results have been computed for 36 time iterations. We have used the and norm to measure the error.

and represents the sum of the and error over the time steps.

Number of elements X-FEM FEM, stationnary mesh
 
20 0.65 1.73 18.34 27.79
100     17.82 25.43
200     6.69 19.33
400     2.64 18.34
In the case of the classical FEM, with 10 elements and 11 nodes, this method exhibit poor accuracy in computing the solution near the front location, which is not surprising as the local feature is contained within a single element (see Fig. 12). It is only with a dramatic increase in the number of element that the computed solution begin to satisfy (see Fig. 13).

Figure 12: The finite element (10 elements) and exact solution.
Figure 13: The finite element (400 elements) and exact solution.

Figure 14: The X-FEM (10 elements) and exact solution.


next up previous contents
Next: Second Problem: the shock Up: First Problem: A localized, Previous: The research of the   Contents
Renaud Merle 2000-08-26