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11

The Finite Element Method (FEM) is the parent method which has inspired many, many other methods and methods which are actually FEM but pretend not to be. In the finite element method, "shape functions" are used to provide an approximation space so that the solution can be represented by a vector. In the classical FEM, these shape functions are polynomials. ...

8

deal.II keeps the entire coarse mesh on every processor, but of course we cannot do that with the actual mesh after many refinement steps. It is true that this somewhat limits the size of problems we can solve to maybe a few hundred thousand coarse mesh cells on typically-sized cluster nodes. However, this is plenty for most realistic cases for a reasonable ...

8

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. Limiting considerations to elliptic problems. The approximate solution can converge to exact solution if you do h- or p- adaptivity. If the exact solution is ...

7

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 squares), and (ii) have local symmetries. Symmetries in a quadrilateral mesh would, for example, mean that four cells come together at every vertex where the four ...

7

Both Mike's answer and Jed's one describe well the XFEM/FEM dichotomy and correctly point out that the most important area of application is 3D Fracture Mechanics, where you have a crack, i.e. a displacement discontinuity across a surface inside your domain. Cracks are hard to model in classical FEM for two reasons: The mesh has to be congruent across the ...

6

For the how part referred to in the previous answer, conforming Quad or Hex mesh refinement is most likely going to use an algorithm based on the work of R. Schneiders' 2- and 3- refinement algorithms. These methods are used in mesh generation. Two papers that I happen to have that do adaptive conforming quad refinement are: "A new fast hybrid adaptive grid ...

6

Every major class of discretization is "open-ended" in the sense that there are decisions with no obviously/provably correct answer in the general case, so some decisions are made based on how they perform for the target problem. Additionally, each major class has active research on new extensions. AMR has more choices than static-grid discretizations, so ...

6

FEM is a subset of XFEM. XFEM is a methodology for enriching finite-element spaces to handle problems with discontinuities (such as fracture). With classical FEM, attaining similar accuracy typically requires complicated conformal meshing and adaptive refinement, where as XFEM does it with a single mesh, moving that geometric complexity into the elements (...

5

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 can parse the output. It also has a C++ API, and the mailing list gets enough traffic (probably 10-20 messages a week) that your questions might be answered (in ...

5

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 following preprint: R. Becker, R. Rannacher: "A feed-back approach to error control in finite element methods: basic analysis and examples". IWR preprint, ...

5

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 website (which also has her code and given that its a collaboration between her, Randy and others), seems like a good start. https://cs.nyu.edu/berger/

4

An adaptive grid is a grid network that automatically clusters grid points in regions of high flow field gradients; it uses the solution of the flow field properties to locate the grid points in the physical plane. The adaptive grid evolves in steps of time in conjunction with a time-dependent solution of the governing flow field equations, which computes ...

4

This would call for adaptive finite element methods, and if you are only interested in specific regions, for the use of goal oriented error estimators to drive the adaptive mesh refinement. The problem you want to solve is a pretty standard one and you can find my own contribution to this in the step-6 tutorial program of the deal.II library for the general ...

4

Since you are a computer science major, let me posit the following analogy: "adaptive mesh refinement" is a set of techniques for solving partial differential equations in mathematics; this is in the same spirit as "image processing" is a set of techniques to transform and improve images. Both fields have many different aspects, so there are no fixed ...

4

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 starting mesh. It doesn't have to be a square. A few examples of adaptive meshes on other domains are here: https://github.com/dealii/dealii/wiki/Gallery 2/ Cell-...

4

If you want the simplest possible numerical scheme working for Burger's equation that has your suggested form then you should prefer the so called Lax-Friedrichs method. If you have the book of LeVeque on Finite Volume Methods for Hyperbolic Problems, look for a very simple formula 4.20 (or a little bit more complex 4.21, but in your suggested form). With ...

4

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 functions, i.e., $(\nabla\cdot\mathbf v_0,q)=0$ for all $q\in P_1(T_0)$ where $T_0$ is the original mesh. What you want to achieve is to get a velocity field $\mathbf ... 4 You may want to take a look at the arXiv preprint of B. Keith, A. V. Astaneh, and L. Demkowicz, "Goal-oriented adaptive mesh refinement for non-symmetric functional setting." In this article, the authors motivate and present a new duality theory for FEM. Some overview of dual-weighted residual is also given (Section 5). It also links to the very widely ... 3 It is difficult to assess the correctness of a code for only a single value of the mesh sizes$h$,$H$. Rather, one typically evaluates the accuracy for a sequence of mesh sizes$h,H \rightarrow 0\$ by comparing against the exact solution to which your numerical approximations have to converge.

3

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 you mentioned would be too refine the top percentage of the elements with greatest error and then also coarsen the bottom percentage of elements with lowest error....

3

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: for (cell in cells) if (cell->is_marked_for_refinement && cell->edge_to_split == unassigned) { e = longest edge of cell (0...2); cell->...

3

@Peter Frolkovic's answer is a good one, but @Daniel Ruprecht's comment also deserves to be highlighted: the scheme you are using (centered in space, forward in time) is unstable for any time step size. It's straightforward to see this if you consider instead the advection equation and do a standard von Neumann or method of lines stability analysis. This ...

3

Here a most likely incomplete list. But maybe others can help extend it. Bisection methods can be applied to simplicial meshes and have the advantage that they are always conforming. Refinement into similar objects can be applied to triangles, quadrilaterals and hexahedra. To tetrahedra with some modifications. It allows for particularly simple setup of ...

3

I was (still am) looking for good answers for this. I work with multi-level adaptive grids where I use some sort of criterion for refinement. Folks doing FEM enjoy, rather cheap (computationally), rigorous error estimates that they use as refinement criterion. For us doing FDM/FVM, I have not had luck finding any such estimates. In this context, if you want ...

3

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 more detail in lecture 15 at http://www.math.tamu.edu/~bangerth/videos.html).

2

If it is indeed 1D then you probably won't need any adaptive mesh here, for such a simple problem you can probably resolve all you need with a static grid, with a computing power of a modern workstation. But it is a perfectly reasonable strategy, in the process of time-integration, to identify periodically areas where the numerical resolution is stressed, ...

2

There is a paper from the 1990s (early 2000s) by Roland Becker on the topic. I think it was co-authored by Rolf Rannacher. There is also a more recent paper by Rannacher and Vihharev that I proof read last year. It may not have appeared, but you could ask them whether they're willing to send you a copy. It is not this one http://numerik.iwr.uni-...

2

I didn't read this in detail, but I think it should have the gist of the kind of proof you're looking for.

2

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. For example my Node and Tree classes look like: template <class T> class Node { public: T element; Node *parent; Node *left; Node *right; ...

2

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 on the first mesh) as evidenced by the fact that your residual is nonzero there, then that will produce an error elsewhere as well. As a consequence, it is ...

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