Laplace Demo 1

Laplace Demo 1#

keys: homogeneous Dirichlet bvp, single layer potential ansatz, single layer potential operator, electrostatics

from netgen.occ import *
from ngsolve import *
from ngsolve.webgui import Draw
from ngbem import *
from ngsolve import Projector, Preconditioner
from ngsolve.krylovspace import CG

Loading ngbem library

Dirichlet Boundary Value Problem

Single Layer Potential

Variational Formulation

\( \left\{ \begin{array}{rcl l} -\Delta u &=& 0, \quad &\Omega \\ \gamma_0 u&=& u_0, \quad &\Gamma \end{array} \right. \)

\(\quad \Rightarrow \quad\)

\( u(x) = \mathrm{SL}(j) \)

\(\quad \Rightarrow \quad\)

\(\left\langle \gamma_0 \left(\mathrm{SL}(j)\right), v \right\rangle_{-\frac12} = \left\langle u_0, v\right\rangle_{-\frac12} \)

\(\mathrm{V} \, \mathrm{j} = \mathrm{M} \, \mathrm{u}_0 \)

NG-BEM Python interface

symbol

FE trial space

FE test space

SingleLayerPotentialOperator

\(\mathrm V \)

SurfaceL2

SurfaceL2

DoubleLayerPotentialOperator

\(\mathrm K \)

\(\gamma_0\) H1

SurfaceL2

HypersingularOperator      

\(\mathrm D\)

\(\gamma_0\) H1

\(\gamma_0\) H1

DoubleLayerPotentialOperator

\(\mathrm K'\)

SurfaceL2

\(\gamma_0\) H1

Mesh

sp = Sphere( (0,0,0), 1)
mesh = Mesh( OCCGeometry(sp).GenerateMesh(maxh=0.3)).Curve(3)
Draw(mesh);

Trial and Test Functions

fesL2 = SurfaceL2(mesh, order=3, dual_mapping=True)
u,v = fesL2.TnT()

Right Hand Side \(\;\mathrm{M}\mathrm{u}_0\)

u0 = 1/ sqrt( (x-1)**2 + (y-1)**2 + (z-1)**2 )
Mu0 = LinearForm (u0*v.Trace()*ds(bonus_intorder=3)).Assemble()

System Matrix \( \; \mathrm{V}\)

V = SingleLayerPotentialOperator(fesL2, intorder=10, method="aca")

Solve \(\; \mathrm{V}\mathrm{j} = \mathrm{M}\mathrm{u}_0\)

j = GridFunction(fesL2)
pre = BilinearForm(u*v*ds, diagonal=True).Assemble().mat.Inverse()
with TaskManager(pajetrace=1000*1000*1000):  
    CG(mat = V.mat, pre=pre, rhs = Mu0.vec, sol=j.vec, tol=1e-8, maxsteps=200, initialize=False, printrates=False)
Draw (j);

Notes: