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

Younghigh
Topic Author
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Junior Member

3 years 2 weeks ago #3632 by Younghigh

About the facet integral in the DG methods for solving the Stokes equation was created by Younghigh

To solve the Stokes equation, we use the DG methods. And the following formula is an important facet integral:

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

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
[tex]\left(
\begin{array}{cc}
a & c \\
b & d \\
\end{array}
\right),[/tex]
nor
[tex]\left(
\begin{array}{cc}
a & b \\
c & d \\
\end{array}
\right).[/tex]

Please help me, thank u.

Best wishes,
Di Yang

hvwahl
Offline
Premium Member

3 years 2 weeks ago #3634 by hvwahl

Replied by hvwahl on topic About the facet integral in the DG methods for solving the Stokes equation

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

cwinters
Offline
Administrator

3 years 2 weeks ago #3638 by cwinters

Replied by cwinters on topic About the facet integral in the DG methods for solving the Stokes equation

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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