Hi

I'm looking for nice 3D benchmark problem for solution of Laplace equation, or Poisson. I would like to compare my Boundary Element Method results with Finite Element Method or Finite Volume Method... and hopefully to beat FEM and FVM... Thank you very much... Matjaz

I have a nice test case but, unfortunately, it has:

* a diffusion coefficient which varies with space

* a non-linear source term

* requires a large of number of elements to resolve the geometry

which, I guess, will be a problem for your BEM but causes no problem for my FEM and FVM schemes.

Seriously though, pretty much any test case works for Laplaces equation. A square box with Dirichlet boundary conditions of 1.0 on one side and 0.0 on the others is about the simplest and as good as any. Could use Neumann and Robin conditions as further tests. Just make sure you compare against a properly working multigrid solver for your FEM and FVM codes.

More meaningful tests would depend on what types of real world problems you want to use the BEM approach to solve. Unfortunately, BEM is only applicable to small parts of CFD. At a guess, heat transfer would seem your best bet for real world test cases (so long as they are not too big).

What do you mean by "beating" FEM/FVM in your tests? In terms of accuracy or cost, or both? A cursory check of the literature on BEM will show you that this has been done many times over in the past by many in the BEM community. There is no doubt, at least in my mind, that BEM can give you as accurate as or better results. That accuracy can/will go down, though, if you apply a FMM to reduce the computational complexity to an order that will make BEM competitive.

The simple problem of a square with Dirichlet or Neumann BC, is a nice benchmark. However, you have to make sure that your boundary condition distribution function is at least one order higher than the shape function you use for the potential (and/or flux) on the boundary. The reason is that if you apply a BC with the same order as your potential distribution, you _will_ get an exact solution to machine precision, as there are no errors anywhere, unlike FEM (that is, assuming your integrals are evaluated in analytic form). For example, if your inlet BC is a constant, and you use piecewise constant shape functions you will get an exact solution. But if you use a parabolic inlet distribution with a piecewise linear shape function, then there is going to be errors, which will allow you to perform a convergence analysis.

Adrin Gharakhani

Andy, thank you for your quick response. Of course, "beating" FEM/FVM by BEM is in terms of accuracy (and to heat you up!). Additional, in terms of cost, my BEM isn't far behind. I'm using multidomain BEM, which means, the domain is discretised similar as in FEM, sparse matrices ... more you can see in http://top25.sciencedirect.com/?journal_id=09557997 the first, leading article titled: "A multidomain boundary element method for ..." In this article, a 2D BEM is used for 500.000 elements, computed in few minutes. In this paper, the diriclet problem you mentioned in square is solved and copmpared. In present 3D problem, I can solve 150.000 elements, variable diffusivity and noonlinear source terms. Please send me your test problem together with results. I hope I will be able to solve it in the middle of September, because I'm out of my office in August. Thank you very much, once more... Greeting from Slovenia, Matjaz

I am afraid Science Direct is giving me an error. I will try again later. Wretched people buy up everything, charge everybody and then, at least here, provide an unreliable service.

Why do you want a Laplace or Poisson test problem? I would have thought a laminar CFD problem would be more appropriate? The most common being the lid driven cavity problem. I see from the abstract (which I can read) you have done the backward facing step and flow past a cylinder.

I do not have a test problem which I can extract easily because Poisson problems in CFD codes tend to be one step of a sequence. If you are interested in non-negligible effort then we would be better discussing it further via email.

I have an interest in dividing down a BEM solution region although not in the area of CFD. It is not apparent to me where the sweet spot is for trading reduced accuracy from the division against increased efficiency from the sparseness. Also, the order of the elements is likely to be play a different role when one has only a handful per sub domain. Has this been part of you study?

You can find the article from my homepage simple typing matjaz ramsak homepage in the google search field and 1st hit is mine... You will find answers in this articles.

In 3D, I have only laplace-poison eq. solver. For CFD, It's not so straight forward using BEM...

I'm not sure what you have in mind, or what your code can handle.

Can you calculate the lift of a flat wing? There is an exact (well, a series solution) for the lift-slope of an infinitely-thin flat wing of circular planform. I have seen published values correct to 7 or 8 decimal places.