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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. ...
likask's user avatar
  • 906
6 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 ...
EMP's user avatar
  • 2,089
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 ...
Wolfgang Bangerth's user avatar
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 ...
Wolfgang Bangerth's user avatar
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 ...
Wolfgang Bangerth's user avatar
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 ...
Wolfgang Bangerth's user avatar
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 ...
Anton Menshov's user avatar
  • 8,672
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 ...
Wolfgang Bangerth's user avatar
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 ...
Wolfgang Bangerth's user avatar
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-...
Wolfgang Bangerth's user avatar
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 ...
Sardine's user avatar
  • 378
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 ...
Daniel Shapero's user avatar
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 ...
Wolfgang Bangerth's user avatar
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\|...
knl's user avatar
  • 2,104
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 ...
Wolfgang Bangerth's user avatar
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. $...
Abdullah Ali Sivas's user avatar
2 votes

simple example of an adaptive mesh refinement code

Newer versions of Clawpack and AMRClaw/GeoClaw do include some 1D versions but I am not sure that's going to be entirely helpful. Donna Calhoun has a page that attempts to list some well-used codes ...
Kyle Mandli's user avatar
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 ...
Guillermo BCN's user avatar
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 ...
cfdlab's user avatar
  • 3,028
1 vote

Recovery of smoothed continuous stresses using the Z^2 error estimator

I just implemented this method as detailed in the book by Zienkiewicz and Zhu (The Finite Element Method - Its Basis and Fundamentals) for linear tetrahedral elements. This is an excellent book, but ...
Charlie S's user avatar
  • 661
1 vote

Data structures of AMR(Adaptive Mesh Refinement) with quadtree

Since the quad-tree has logarithmic height, the overall space cost is small. If your mesh were the same everywhere, you could do implicit packing, as in a heap. For a binary tree, that looks like ...
Richard's user avatar
  • 3,971
1 vote

How can I make sure the flow is divergence-free when I use moving mesh?

What sort of field is it? You've said "divergence free", but do you mean harmonic (zero divergence and curl), solenoidal (non-curl), or a mix of both? The distinction is important because it will ...
Sean Lake's user avatar
  • 143
1 vote

fast adaptive quadrature on equispaced 2-D grid

I don't think it's quite clear to me why an adaptive scheme would be better. Integration with the trapezoidal rule is very cheap: It's in essence just one addition and one multiplication per grid ...
Wolfgang Bangerth's user avatar

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