# About the facet integral in the DG methods for solving the Stokes equation

3 years 5 months ago #3632
To solve the Stokes equation, we use the DG methods. And the following formula is an important facet integral:

$$\sum_{F\in\mathcal F_h}\oint_F\left\{\nabla\bm{u}\right\}\bm{n}_F\cdot \left[\bm{v}\right]\mathrm{d}F.$$

If coding in the NGSolve, I try this:
Code:
def outer(W,n): return CoefficientFunction((W[0]*n[0], W[0]*n[1], W[1]*n[0], W[1]*n[1]), dims=(2,2)) def grad(a,b): return CoefficientFunction((a.Diff(x), b.Diff(x), a.Diff(y), b.Diff(y)), dims=(2,2)) fes = L2(mesh, order=order_p, dim=2, dgjumps=True) U, V = fes.TnT() ux, uy = U vx, vy = V n = specialcf.normal(2) ux_other, uy_other = U.Other() vx_other, vy_other = V.Other() U_vec = CoefficientFunction((ux, uy)) U_Other = CoefficientFunction((ux_other, uy_other)) V_vec = CoefficientFunction((vx, vy)) V_Other = CoefficientFunction((vx_other, vy_other)) jump_U = U_vec - U_Other jump_V = V_vec - V_Other mean_gradU = 0.5*(grad(ux, uy) + grad(ux_other, uy_other)) mean_gradV = 0.5*(grad(vx, vy) + grad(vx_other, vy_other)) a = BilinearForm(fes) a += SymbolicBFI( InnerProduct( outer(jump_V, n), mean_gradU ),\ VOL, skeleton=True ) a += SymbolicBFI( InnerProduct( outer(V_vec, n), grad(ux,uy) ),\ BND, skeleton=True )

I cannot sure it is correct because I have no idea that the normal
Code:
n=specialcf.normal(2)
is a column vector nor a row vector.

In addition, if one codes like
Code:
CoefficientFunction((a,b,c,d), dims=(2,2))
the corresponding matrix is
$$\left( \begin{array}{cc} a & c \\ b & d \\ \end{array} \right),$$
nor
$$\left( \begin{array}{cc} a  & b \\ c & d \\ \end{array} \right).$$

Best wishes,
Di Yang
3 years 5 months ago #3634 by hvwahl
Hi Di Yang,

I think that the problem is that you're using the wrong FESpace. VectorL2 should be the one you need:
Code:
from ngsolve import * from netgen.geom2d import unit_square mesh = Mesh(unit_square.GenerateMesh(maxh=0.1)) V = VectorL2(mesh, order=3, dgjumps=True) u, v = V.TnT() n = specialcf.normal(mesh.dim) jump_u = u - u.Other() jump_v = v - v.Other() mean_grad_u = 0.5 * (Grad(u) - Grad(u.Other())) mean_grad_v = 0.5 * (Grad(v) - Grad(v.Other())) a = BilinearForm(V) a += InnerProduct(mean_grad_u * n, jump_v) * dx(skeleton=True) a += InnerProduct(Grad(u) * n, v) * ds(skeleton=True)

Best wishes,
Henry
The following user(s) said Thank You: Younghigh
3 years 5 months ago #3638 by cwinters
Hi Di Yang,

concerning your question about coefficient functions, the input is taken row-wise. For
Code:
cf = CoefficientFunction((1,2,3,4), dims=(2,2))
you can get out the components by
Code:
cf[i,j]
and I think evaluating (or drawing) the components is probably the quickest way to check the alignment.
Assume you have a mesh of the unit square, you could call
Code:
cf[0,1](mesh(0.5,0.5))
to get the output 2.

Best,
Christoph
The following user(s) said Thank You: Younghigh
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