This page was generated from unit-11.2-bem-Laplace/Laplace_DtN_indirect.ipynb.

11.2.1 Dirichlet Laplace Indirect MethodΒΆ

keys: homogeneous Dirichlet bvp, single layer potential, ACA

[1]:

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

We consider the Dirichlet boundary value problem

\[\begin{split}\left\{ \begin{array}{rcl l} \Delta u &=& 0, \quad &\Omega \subset \mathbb R^3\,,\\ \gamma_0 u&=& u_0, \quad &\Gamma = \partial \Omega\,.\end{array} \right.\end{split}\]

Let us choose the following ansatz for the solution \(u\in H^1(\Omega)\) (indirect ansatz)

\[u(x) = \underbrace{ \int\limits_\Gamma \displaystyle{\frac{1}{4\,\pi}\, \frac{1}{\| x-y\|} } \, j(y)\, \mathrm{d}\sigma_y }_{\displaystyle{ \mathrm{SL}(j) } }\]

and solve for the density \(j\in H^{-\frac12}(\Gamma)\) by the boundary element method, i.e. the numerical solution of the variational formulation

\[\forall \, v\in H^{-\frac12}(\Gamma): \quad \left\langle \gamma_0 \left(\mathrm{SL}(j)\right), v \right\rangle_{-\frac12} = \left\langle u_0, v\right\rangle_{-\frac12} \,.\]

Define the geometry \(\Omega \subset \mathbb R^3\) and create a mesh:

[2]:

sp = Sphere( (0,0,0), 1)
mesh = Mesh( OCCGeometry(sp).GenerateMesh(maxh=0.3)).Curve(4)

Create test and trial function finite element spaces for \(H^{-\frac12}(\Gamma)\) according to the given mesh:

[3]:

fesL2 = SurfaceL2(mesh, order=4, dual_mapping=True)
u,v = fesL2.TnT()

Define Dirichlet data \(u_0\) and compute the right hand side vector:

[4]:

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

The discretisation of the above variational formulation leads to a system of linear equations, ie

\[\mathrm{V} \, \mathrm{j} = \mathrm{rhs} \,,\]

where \(\mathrm{V}\) is the single layer potential operator. \(\mathrm V\) is regular and symmetric.

Demo 1: Assemble the single layer operator \(V\) as dense matrix and solve for unknwon density \(j\):

[5]:

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

CG iteration 1, residual = 2.185606954637756
CG iteration 2, residual = 0.551099029879933
CG iteration 3, residual = 0.13112741190747146
CG iteration 4, residual = 0.031078921186674928
CG iteration 5, residual = 0.0073724818546740695
CG iteration 6, residual = 0.0017929371818998655
CG iteration 7, residual = 0.0005259807041763909
CG iteration 8, residual = 0.0002580783579649913
CG iteration 9, residual = 0.0002830928616280549
CG iteration 10, residual = 0.00011690091777021659
CG iteration 11, residual = 0.00021437370035804374
CG iteration 12, residual = 4.5279381469980324e-05
CG iteration 13, residual = 4.801536436570242e-05
CG iteration 14, residual = 2.782589491318866e-05
CG iteration 15, residual = 1.4630501833773815e-05
CG iteration 16, residual = 1.4784232981525944e-05
CG iteration 17, residual = 9.772740421352912e-06
CG iteration 18, residual = 6.0800277817262885e-06
CG iteration 19, residual = 1.4674974806460037e-05
CG iteration 20, residual = 3.302657376399434e-06
CG iteration 21, residual = 2.9696997933417716e-06
CG iteration 22, residual = 3.492904504633916e-06
CG iteration 23, residual = 2.5938240703176614e-06
CG iteration 24, residual = 1.2605882370469452e-06
CG iteration 25, residual = 7.886119679737291e-07
CG iteration 26, residual = 1.4658939487284313e-06
CG iteration 27, residual = 5.877586698591265e-07
CG iteration 28, residual = 4.4334941709822883e-07
CG iteration 29, residual = 5.853360472952393e-07
CG iteration 30, residual = 3.510906343312404e-07
CG iteration 31, residual = 1.7794563636211842e-07
CG iteration 32, residual = 1.5348852818976806e-07
CG iteration 33, residual = 2.2021949908818368e-07
CG iteration 34, residual = 1.2615655749479873e-07
CG iteration 35, residual = 8.735320499061673e-08
CG iteration 36, residual = 7.029355521372671e-08
CG iteration 37, residual = 9.921227489332353e-08
CG iteration 38, residual = 3.434517171191766e-08
CG iteration 39, residual = 4.059008716696599e-08
CG iteration 40, residual = 5.6418830192562196e-08
CG iteration 41, residual = 1.8555959315079995e-08

[6]:

Draw (j, order=3);

Notes:

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