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Astrid Barros November 12, 1998 20:55

Gradient Matrix in FEM
At the moment I am developing a FEM code to solve the Navier Stokes equations splitting the equation in three. The equations are: one for convection, one for diffusion and one for pressure. In the pressure equation I am obtaining an anti-simetric matrix. I would like to know if someone has had such experience and any idea of where could be the bug. I really would be glad to have some sugestions......!

Astrid Barros November 18, 1998 20:25

Re: Gradient Matrix in FEM
Hello ThoLi,

I have tried working with two equations also: Burger's eq. for velocity and Poisson eq. for pressure,

using the projection method (Chorin), and also I was using analitycal integration for the Galerkin integrals.

Some strange things were hapenning, the solution started converging to a very good solution and instead of converging

(small variations in velocity) it continued incrementing, diverging from the desired solution.


And so I did the splitting in three eqs., in a way that I could verify the evolution of each one, and I also used Gauss Quadrature to do the integration. The simetric matrices are OK, mass: Ni Nj and grad: dNi/dx dNj/dx, x = x, y,

but the other ones become antissimetric, for example, the divergent: Ni dNj/dx, x = x,y, and the non-linear convective term has a similar problem: Ni dNj/dx Nk.


For example, if I have flow over a plate, it is ok, if I have flow in a duct, one side will be the opposite to the other.


If you could send me na example of a divergent matrix, it would be very usefull for me to find my error, because I haven't found this matrix in any literature or papers.


Thanking you very much for your interest and help,

Sincerely yours,




*My equations:

term1: viscosity (second order derivatives of velocity) - Equation 3

term2: convection (first order derivates) - Equation 1

term3: gradient of pressure Equation 2: two steps: grad(P) = div(v) / dt

term4: divergence of velocity dv = - div(p) * dt

ThoLi November 23, 1998 13:25

Re: Gradient Matrix in FEM
Hello Astrid,

please send me (or post) also the following informations: which finite elements do you use (shape: triangular/rectangular; linear,quadratic,... polynomials; integration gaussian (how many points?); )

What variables are involved? (v: velocity?; p: pressure? P: also pressure? t: time? are there more? )

How big is your Reynolds-number?

Have you tried zero convection (= zero Reynolds)?

Describe your time-discretization scheme! (What have you done with dv/dt? with div(v)/dt?)

give me some sketches of your geometry! and of the flow directions, you expect! you can send me a fax at +49 89 - 63812-515 (it's a shared fax, so send also a page it's for me!)


I used:

rectangles with bi-quadratic polynomials, i.e. x^2 * y^2 was included;

integration with a four-point-gaussian in x- and y-direction (resulting in 16 = 4*4 points)

v: velocity, (x,y) -> (vx,vy) ; p: pressure, (x,y) -> p

(no time, no time-discretization)

My Reynolds-Number was 20; I had an exact solution for the case Reynolds=0 and incremented the Reynolds-number (the influence of the convection) in small steps;

I tried SIMPLE as solving algorithm, but then used a direct solver, because I didn't find the right parameters for SIMPLE to converge ...

I'll send you a matrix, when I know your elements ...

greetings ThoLi

Astrid Barros November 30, 1998 21:09

Re: Gradient Matrix in FEM
Hello ThoLi, I am writting the formulation I've been using and as soon as it is ready I'll send it to you. Thank you for your interest, Astrid PS: can I send you a Word97 file? or would it be better to fax it?

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