8
votes
Accepted
p-refinement in adaptive methods
The decision whether to do p- or h- adaptivity is to achieve faster, potentially exponential convergence rate. In other words to get a solution with given error with minimal computational effort.
...
- 906
7
votes
Accepted
Finite element results by different meshes
As a general rule, finite element solutions are more accurate on meshes with cells that (i) deviate less from the optimal shape (which for triangles are equilateral triangles and for rectangles are ...
- 54.1k
6
votes
Accepted
Good references for dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)
The canonical "first" reference for the method is a paper by Becker and Rannacher that was ultimately published as an article in the ENUMATH 97 proceedings, but is often cited as the ...
- 54.1k
6
votes
Accepted
Mesh refinement in the Finite Element Method
You really don't want to implement this yourself -- you'll spend a year or two on things others have already done, and will have done far better than you can hope for.
The difficulty is generally ...
- 54.1k
5
votes
Accepted
simple example of an adaptive mesh refinement code
Marsha has made quite a bit of her source code available over the years. Some of it is no longer supported, but given that she is the Berger in Berger and Oliger, checking her website and the clawpack ...
- 2,069
5
votes
Accepted
Finite element mesh software
I'd say Gmsh. I used it for a few finite element projects, and it was mostly easy to work with. The mesh output formats are very parseable, and there's at least one third-party parser (MeshPy) that ...
- 30.3k
4
votes
Accepted
Implementing Finite Difference Adaptive Mesh Refinement code
I'll try to answer your questions one-by-one, though I'm afraid that they are so basic that these answers alone won't be able to help you very much in the long run:
1/ No, one can run AMR on any ...
- 54.1k
4
votes
How can I make sure the flow is divergence-free when I use moving mesh?
It's important to realize that the original velocity field $\mathbf v_0$ is also not divergence free in a pointwise sense. Rather, it is only divergence free when tested with the pressure test ...
- 54.1k
4
votes
Good references for dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)
You may want to take a look at the arXiv preprint of
B. Keith, A. V. Astaneh, and L. Demkowicz, "Goal-oriented adaptive mesh refinement for non-symmetric functional setting."
In this article, the ...
- 8,602
4
votes
Accepted
Connectivity of octree as grid
The one paper you need to know is the one by Burstedde and collaborators on the implementation of the p4est library: p4est: SCALABLE ALGORITHMS FOR PARALLEL ADAPTIVE
MESH REFINEMENT ON FORESTS OF ...
- 54.1k
4
votes
Accepted
numerical integration of integrals in the p-adaptive version of the finite element method
Yes, a typical choice is to use the maximum -- which happens to be based on the polynomial degree of the finite element used for the element (notwithstanding the fact that some of its degrees of ...
- 54.1k
3
votes
Mesh aspect ratio issue with adaptive mesh refinement (AMR)
All "real" implementations of adaptive mesh refinement start from an unstructured coarse mesh because the domains one wants to solve on are just not always rectangles :-) So your CASE 3 is ...
- 54.1k
3
votes
How to implement adaptive mesh refinement using conformal triangles
The way this is typically handled is by running a pass before you actually split cells that determines which other cells also have to be refined. In pseudo-code, this could look like this:
...
- 54.1k
3
votes
Accepted
How to deal with transition elements in adaptive fem
You will need to estimate the error on all cells, including transition cells. You may then wish to refine these differently, if necessary -- see for example the Red-Green Strategy (which I explain in ...
- 54.1k
3
votes
Accepted
Strategies for controlling number of new elements in adaptive mesh refinement
This is not a complete answer but based on my own experience with mesh refinement I felt compelled to write of few ideas/thoughts which would be too long for a comment.
One idea that I don't think ...
- 1,879
3
votes
AMR-Capable meshing software that is not based on quad/octrees
deal.II can subdivide cells in ways that are not geometrically 2:1, but "graded" in certain directions. It does this by "mapping" the new mid-points of edges and cells in non-...
- 54.1k
3
votes
AMR framework for efficient simulation of PDEs, potentially with boundary layers
Let me preface by saying you may have set on a path more adventurous than you had hoped... Mesh generation itself is often a bottleneck, let alone rigorous 4D adaptation of complex geometries powered ...
- 368
2
votes
Accepted
Using finite element error estimators for adaptive mesh refinement
As mentioned in a comment below the question, the formula you state is incorrect: it should read
\begin{equation}
\eta^{2} = \Sigma_{F\epsilon{}\partial{K}}C_{F}\int_{\partial{K}_{F}}([\nabla{u_{h}...
- 54.1k
2
votes
unstructured grid AMR
I ended up with a simple AMR implementation that does the job for polyhedral cells
a) No coarsening. The original coarse grid is always loaded first on which refinement is applied. This gets rid of ...
- 233
2
votes
Accepted
How to implement adaptive mesh refinement using conformal triangles
With the help of Wolfgangs answer I think I have figured it out. It is necessary to include additional pointers in my Node class so that each Node in the binary tree can know who are its neighbors. ...
- 1,879
2
votes
Accepted
Refinement in AMR
Discretizations of partial differential equations "enjoy" a property called error pollution that means that if you don't exactly satisfy the equation at one location (such as the points you identified ...
- 54.1k
2
votes
Accepted
Non-uniform finite difference Adaptive Mesh Refinement
You don't typically form a finite difference stencil involving the equation at these "hanging nodes", but set the value of this vertex to the mean value of the two adjacent vertices along the edge.
(...
- 54.1k
2
votes
Finite element results by different meshes
I have written an exhaustive answer on meshing here: https://engineering.stackexchange.com/questions/449/meshing-of-complex-geometrical-domains/7326#7326
The resultant mesh quality is more important ...
- 21
2
votes
Accepted
Oscillation term in a posteriori error estimator
In the efficiency proof you use bubble functions in order to get rid of the elementwise boundary terms. For a bubble function $b_T$ defined on an element $T$ it holds
$$c \|v_h\|_{0,T} \leq \|b_T v_h\|...
- 2,041
2
votes
Comparison on adaptive mesh refinement on finite elements and finite differences
Finite element methods are generally easier to deal with when using adaptive mesh refinement because higher order finite difference methods have stencils that extend for several mesh sizes away from a ...
- 54.1k
2
votes
Computing the residual in a Dual Weighted Residual (DWR) method
Take my advice with a bit of salt, as I am not an expert on adaptive FEM. I don't have access to the papers, so I am not sure if the following is how they do it, but it is how I would implement it.
$...
- 2,411
2
votes
Accepted
Data structures of AMR(Adaptive Mesh Refinement) with quadtree
Your data structure is fine for a toy problem, but it's not general enough, not efficient, for real applications:
You make the assumption that your mesh consists only of squares and that consequently ...
- 54.1k
2
votes
Adaptive mesh refinement with inter-element continuity
It's been a while since I've used deal.II so this is my recollection.
If Wolfgang Bangerth or one of the other developers says otherwise you should listen to them.
What deal.II does is add continuity ...
- 10.1k
1
vote
Accepted
Proof of R. Verfürth paper on adaptive mesh and bubble functions
He is only going to the reference triangle to have the constant independent of $h$, as required. But yes, he is using the fact that having the bubble function inside the norm double bars does not stop ...
- 126
1
vote
Accepted
Finite Difference libray C++
There are several libraries for adaptive grids, see e.g.,
https://math.boisestate.edu/~calhoun/www_personal/research/amr_software/
I have found Petsc to be very useful to write finite difference ...
- 2,993
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