[petsc-users] Compute integral using DMPlexComputeIntegralFEM

[Matthew Knepley]

On Fri, Sep 8, 2023 at 6:26 PM David Andrs <andrsd at gmail.com> wrote:

> Hi all!
>
> I am trying to use DMPlexComputeIntegralFEM to compute an integral
> $\int_\Omega u d\Omega$. My domain is a square (-1, 1)^2 (2x2 QUAD4
> elements), I add first order Lagrange FE field on it, set the solution
> vector (computed by a previous simulation).
>
> The value I am seeing computed by PETSc is -4, but the hand-calculated
> value of this integral is -4.6. I also checked this in paraview using the
> ‘Integrate Variables’ filter and it also returns -4.6 (this was to double
> check that my hand-calculated value is correct).
>

Sorry it took so long. You caught me at a bad time.

Something must be wrong with your analytic integrals. Here is me doing them
by hand. You have a 3x3 vertex arrangement with coefficients

   1  0 -3
  -2 -1 -2
  -3  0  1

>From the symmetry, the integrals of the cells along each diagonal must be
equal. Now, the shape functions for Q_1 are

     (1 - x) y            x y

  (1 - x)(1 - y)     x (1 - y)

Thus the integral for the lower left cell is

  \int^1_0 dx \int^1_0 dy -3 + 3 x + y - 2 xy = -3 + 3/2 + 1/2 - 2/4 = -1.5

which is also the upper right cell. The integral for the lower right cell is

  \int^1_0 dx \int^1_0 dy x - y - 2 xy = 1/2 - 1/2 - 2/4 = -1/2

which is also the upper left cell. Thus we get -1.5 - 1.5 - 0.5 - 0.5 = -4,
which is what Plex gets.

  THanks,

    Matt

So, I must be missing something obvious in my code. Attached is the minimal
> PETSc code to show what I am doing. This is against PETSc 3.19.4.
>
> Thanks in advance for your help,
>
> David
>
> --
>
>
>

-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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