Narrow band with unfitted element scheme.

The width of band depends on mesh size h. I got following convergence plots:

To try strip width less than 3h, I would need to add some extra code to handle triangles that get cut off twice.

Example 5.1 from Eliott's paper:

$\Delta u = f$ on [-2,2]x[-2,2] \ Unit circle
$u = g_D$ on the sides of the square
${\partial u \over \partial n} = g_N$ on unit circle boundary
with
$f = -r^2 \\
u = (r^2+r^{-2})cos(2 \theta) + {r^4 \over 12}(sin^4(\theta)+cos^4(\theta)) \\
g_N = -(x^4+y^4)/3$


I get O(h^2) convergence in L2 norm:

 

Next: Apply the now working unfitted fem to narrow band

Found an error: My code occasionally subdivided quadrilaterals incorrectly.

Blue: interior
Green: boundary triangles
Red: part of boundary triangles that is actually used

After fixing above problem, I get following convergence plot.

I have also made the grid symmetric wrt axes and removed the code to zero-mean the solution.

Error plots
h = 0.05

 

0.0125

Deckelnik/Eliott method implementation (2)

I was able to significantly improve and speed up my code, which was needed to run more tests with finer meshes:

- Added ILU preconditioner and changed iterative solver from GMRES to CGS (conjugate gradient squared)

- Switched mesh generation to 'dumb' gridding with squares or hexagons. Since we don't actually need a mesh for any specific domain, I generate a very simple grid that contains required region, and restrict to that grid. Another benefit is that all triangles come out exactly same, so I simplified a number of matrix assembly steps by computing stiffness and mass sub-matrices once.

- Profiled my code and removed several bottlenecks. Most important of those was fixing the ways I handled the sparse matrix

This allowed me to run a couple more refinements for last week's results. Unfortunately it looks like I was wrong, I do not have O(h^2) convergence:

Update: I found an error in my reference triangle calculations (item 2 in list above).
Update 2: Above error with indexing only affected square meshes. Equilateral triangles in hex meshes are not affected by this error.

I have also checked again if my overall FEM implementation was ok, using a simple example u=cos(pi x)cos(pi y) on a [-1 1]x[-1 1] square. It was correct:

 

 So, I am still unclear why I did not get the Deckelnik unfitted method to work.

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: