Divergence free polynomials
I want to use divergence-free basis in finite element framework for discretizing the Maxwell equations due to divergence free magnetic field.
All paper which i read, include vector free divergence basis for approximation field. Scalar function for approximation of Hx,Hy distinct, how i will carry out?
 J. T. Holdeman. "A Hermite finite element method for incompressible fluid flow." Int. J. Numer. Meth. Fluids, 64:376-408 (2010)
 J. T. Holdeman. "Computation of incompressible thermal flows using Hermite elements." Comput. Meth. Appl. Mech. Engrg., 199:3297-3304 (2010)
The finite elements described are strictly divergence-free and application is straight-forward. The trick is that the polynomials used are necessarily Hermite. The papers describe applications in 2D, but a paper for 3D is under development and should be available in preprint form within a year. Irrotational fields are even simpler (mentioned very briefly in ) as is extension to 3D.
Thanks. I will read carefully these papers for 2d case. Finally, i must construct the 3d version of maxwell solver and thus i will need solenoidal basis for 3d elements. Initially, i read the paper "Locally divergence-free discontinuous galerkin methods for the Maxwell equations." by Cockburn, Shu and Li and i saw that very easy can be constructed div-free vector bases (i have 2d and 3d vector basis for rect and hexa). What is your opinion for these basis? Have you worked with these?
The velocity elements I work with and described in the papers are the curl of a stream function or vector potential element and hence are strictly divergence-free locally. In addition they have continuous normal components at element boundaries and hence are divergence-free there (tangential continuity is not a requirement). Discontinuous elements are obviously not divergence-free at the element interfaces except perhaps on average. Both yield satisfactory results, but implementation is simpler with the Hermite elements.
The 3D element I referred to is again the curl of a vector potential element and has necessary normal continuity at element interfaces, but some VP degrees-of-freedom that do not contribute to the velocity (curl) have been eliminated to produce a simpler velocity element. That VP form is probably not suitable for vector potential formulations in electromagnetics, but that can be remedied.
The sense of vector div-free basis is to be zero the divergence of each of these. Therefore, independenly of DOFs, the div of approximation is set zero.
Consequently, in case of vector basis, i must use only one set of DOFs for approximation of solution vector field. If i use a set of DOFs for approximation of each solution vector componets(scalar), distinctly, i can not carry out the desirable property. For example:
Hx,Hy are componets of vec(H), ci,gi two sets of DOFs (i=0..n) and Pi,Fi scalar basis.
div(vec(H))=d(Hx)/dx + d(Hy)/dy =Σ ci *d(Pi)/dx + Σ gi *d(Fi)/dy =0 if all Pi=Fi=0 or if d(Pi)/dx=-d(Fi)/dy with special values of DOFs ci,gi.
Otherwise if i use one set of DOFs for approximation of vector field (not each scalar componets) i have:
div(vec(H))=Σ ci*vec(Pi)=ci*Σ div(vec(Pi))=0 for all cases.
The above problem i have when i take as scalar basis for Hx,Hy the componets of vector div-basis which i read in paper of Shu.
Perhaps the simplest solenoidal vector element on a rectangle has 12 DOFs. At each corner node the DOFs are a stream function and two components of the velocity. For magnetics these would be the z-component of the vector potential and Hx and Hy. Each vector shape function for the 12 DOFs is solenoidal. Thus for any choice of coefficients the resulting field on the element will be solenoidal. So you only have to find values for the coefficients which fit your problem.
There are also triangular solenoidal elements.
Perhaps we could continue this discussion offline.
It is necessary to put the potential component? Using only components
(Hx,Hy) as DOFs, can not i carry out the div zero? Having the stream
function as addition DOFs you raise the total number of DOFs.
Also, the expansion (approximation) of each DOF as you say what scalar
basis will be used? the component of vector basis?
Is it necessary to put the potential component?
The answer is yes. Failure to recognize this is, I think, why this type of element was not recognized 25 years ago. The existance of a divergence-free field v necessarily implies the existence of a potential a such that v=curl a.
For 2D fluids this is no problem. The Navier-Stokes equation can be orthogonally decomposed into divergence-free and curl-free parts. The divergence-free part is a governing equation for the velocity independent of pressure. The curl-free part is related to the pressure Poisson equation.
So how do you control flow through a duct, as an example of pressure-driven flow, when there is no pressure in the equation? The answer is by boundary conditions on the stream function, the net flow is equal to the difference in stream function across the duct. Is this additional work? No, because the stream function replaces the pressure.
What about electromagnetics? If you are constructing divergence-free fields, you still must introduce an additional (potential?) function to project out the non-solenoidal part introduced by using uncorrelated scalar functions for Hx & Hy. So there is no additional work, and you don't have to solve a Poisson equation at each step.
I don't have enough experience with 3D solenoidal elements to understand what economies might be involved. But, you are not constrained by elements that satisfy the LBB condition. Things are complicated by the fact that the dimension of the kernel (null space) of the curl operator is relatively large.
Gresho & Sani point out in their 1998 book (p.468) that there is roughly one momentum conservation and one mass conservation equation at each point in the mesh. They argue that roughly the constraint ratio (mass/momentum) should be roughly one for "good" elements. So with solenoidal Hermite elements, the extra 3 VP DOFs and 3 velocity DOFs result in a constraint ratio of one. There is, in fact, no penalty for introducing the VP DOFs.
Again, in electromagnetics, you don't have a pressure but you must introduce an extraneous potential function to satisfy the divergence-free constraint.
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