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9.2 Implement our own system assembling

In this tutorial we

• write integrators for \(\int_T f v dx\) and \(\int_T \nabla u \nabla v dx\): myIntegrator.hpp myIntegrator.cpp

• put together element matrices to the global system matrix: myAssembling.cpp

from ngsolve import *
import pyngcore   # for timers

from ngsolve.webgui import Draw
from netgen.occ import unit_square

from ngsolve.fem import CompilePythonModule
from pathlib import Path

m = CompilePythonModule(Path('myassemblemodule.cpp'), init_function_name='mymodule')
dir (m)

['MyAssembleMatrix',
 'MyLaplace',
 'MySource',
 '__doc__',
 '__loader__',
 '__name__',
 '__package__',
 '__spec__']

mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))
fes = H1(mesh, order=1, dirichlet=".*")
u, v = fes.TnT()

use our own integrators for element matrix calculation:

a = BilinearForm(fes)
a += m.MyLaplace(1)
a.Assemble()

f = LinearForm(fes)
f += m.MySource(x)
f.Assemble();

gfu = GridFunction(fes)
gfu.vec.data = a.mat.Inverse(fes.FreeDofs()) * f.vec
Draw (gfu);

we can inspect the element matrix:

mylap = m.MyLaplace(1)
ei = ElementId(VOL,17)
mylap.CalcElementMatrix(fes.GetFE(ei), mesh.GetTrafo(ei))

 0.665986 -0.426433 -0.239553
 -0.426433 0.648429 -0.221997
 -0.239553 -0.221997 0.46155

use our own matrix assembling function:

for l in range(4): mesh.Refine()
fes.Update()
gfu.Update()
f.Assemble();

pyngcore.ResetTimers()
print ("num elements =", mesh.ne, ", ndof =", fes.ndof)
with TaskManager(pajetrace=10**8):
    # using our integrator
    mymatrix = m.MyAssembleMatrix(fes, m.MyLaplace(CF(1)), parallel=False)

    # using NGSolve built-in symbolic integrator
    # mymatrix = m.MyAssembleMatrix(fes, (grad(u)*grad(v)*dx)[0].MakeBFI(), parallel=True)

# print ("my matrix = ", mymat)

if fes.ndof < 100000:
    gfu.vec.data = mymatrix.Inverse(fes.FreeDofs()) * f.vec
    Draw (gfu);

num elements = 14336 , ndof = 7329
We assemble matrix
sequential assembling

Compare timings for

• MyLaplace and grad(u)*grad(v)*dx

• sequential and parallel

• for different polynomial orders

for t in pyngcore.Timers():
    if t["name"].startswith("MyAssemble"):
        print (t["name"],"   ", t["time"])

MyAssemble - calc element matrix     0.009387248697733728
MyAssemble - assemble matrix     0.015653053439919278
MyAssemble - buildmatrixgraph     0.0035093344664998503

Exercise:

• Implement MyAssembleVector for building the right hand side vector for \(\int_\Omega f v \, dx\)

• Implement MyNeumannIntegrator for evaluating \(\int_{\partial \Omega} g v \, ds\)