Deckelnik-Eliott unfitted scheme tests

 I tested the unfitted FE scheme from Deckelnik paper on a unit circle, solving 

$-\Delta u + u = f$
on unit circle,

& Neumann BC

for a known
$u = cos(\pi r^2)$

Here are my error convergence results:

It appears there is no problems with my implementation.

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Unfitted finite elements approach from Deckelnick / Eliott / Dziuk paper


imajna.oxfordjournals.org/cgi/reprint/drn049v1.pdf

This paper describes almost the same approach that we are using - obtaining solution on surface by extending to unfitted mesh containing h-narrow band around the surface.

Only difference seems that they are using orthogonal projection from earlier Dziuk's paper to extend the domain, and they do mention Greer's idea as well.

In essence, the unfitted mesh is restricted to a fitted mesh. Elements that fall across the boundary level sets are chopped to only the part inside the band. This creates problems when the restricted triangle comes out very small or narrow.














In the paper, diagonal preconditioning is used. Here are my convergence results with no preconditioning:

       

L2 convergence of restrictions to bands around surface

1. We expected the L2 error on Surface to level off with increasing width of band after a certain value. The following plot confirms this:

(The series represent meshes of edge sizes 0.002, 0.005, 0.01 restricted to bands of increasing width around the surface)

2. Next I refine the mesh and keep same strip width from above runs.

 

Conclusion:

While it is true that sufficient strip width (0.5) gives O(1/nodes) convergence [same as O(1/h^2)], it seems discouraging as we have to take quite a number of triangles that are not crossed by the surface. Figure below shows that the jagged boundary causes the method to behave worse than the case with simple square extension.  am currently looking for ways to improve this, as well as searching how did other people handle such problems.

Related papers I am looking at:

http://www.sciencedirect.com/science/article/pii/0898122185900586#

The problem is to solve a 2D elliptic equation with moving interface using a fixed unfitted mesh. Vertical derivatives of solution across the interface are prescribed.

 

Their approach is also to split the interface gradient into 2 components, one of which is given by oblique condition and the other component is added to the equation matrix. The ellipticity of the problem is proved.