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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 = SingleLayerPotentialOperator(fesL2, intorder=10)
    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 = 2.185606954606291
CG iteration 2, residual = 0.5510990298605162
CG iteration 3, residual = 0.13112741187311436
CG iteration 4, residual = 0.031078921260809356
CG iteration 5, residual = 0.00737248197512427
CG iteration 6, residual = 0.0017929371359740656
CG iteration 7, residual = 0.0005259800974227873
CG iteration 8, residual = 0.00025807777882157614
CG iteration 9, residual = 0.0002832379507301255
CG iteration 10, residual = 0.00012036650171289751
CG iteration 11, residual = 0.0002350721269187123
CG iteration 12, residual = 4.492369119640061e-05
CG iteration 13, residual = 4.801474719098149e-05
CG iteration 14, residual = 2.782522946270136e-05
CG iteration 15, residual = 1.4627805195034785e-05
CG iteration 16, residual = 1.4781005107939614e-05
CG iteration 17, residual = 9.77155502584065e-06
CG iteration 18, residual = 6.175956852560619e-06
CG iteration 19, residual = 1.5387388938641656e-05
CG iteration 20, residual = 3.2801499859050853e-06
CG iteration 21, residual = 2.9700624426539455e-06
CG iteration 22, residual = 3.4931486364538537e-06
CG iteration 23, residual = 2.593794375425068e-06
CG iteration 24, residual = 1.2604006701717951e-06
CG iteration 25, residual = 7.800941052109296e-07
CG iteration 26, residual = 1.5231751958190097e-06
CG iteration 27, residual = 6.205158938543229e-07
CG iteration 28, residual = 7.443526774580593e-07
CG iteration 29, residual = 4.2552187102150487e-07
CG iteration 30, residual = 3.3410972783364483e-07
CG iteration 31, residual = 1.7732772187384787e-07
CG iteration 32, residual = 1.482366755773741e-07
CG iteration 33, residual = 2.084289238278042e-07
CG iteration 34, residual = 1.3173634275085578e-07
CG iteration 35, residual = 9.537116432477149e-08
CG iteration 36, residual = 1.2705350194149222e-07
CG iteration 37, residual = 6.22042375520104e-08
CG iteration 38, residual = 3.4236067256966696e-08
CG iteration 39, residual = 3.6784293866455033e-08
CG iteration 40, residual = 6.992657982846363e-08
CG iteration 41, residual = 1.8642085021274685e-08

[6]:

Draw (j, order=3);

Notes:

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