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Adaptivity

from ngsolve import *
from ngsolve.webgui import Draw

Define the geometry by 2D Netgen-OpenCascade modeling (new in Netgen/NGSolve 2105):

from netgen.occ import *
from netgen.webgui import Draw as DrawGeo

def MakeGeometryOCC():
    base = Rectangle(1, 0.6).Face()
    chip = MoveTo(0.5,0.15).Line(0.15,0.15).Line(-0.15,0.15).Line(-0.15,-0.15).Close().Face()
    top = MoveTo(0.2,0.6).Rectangle(0.6,0.2).Face()
    base -= chip

    base.faces.name="base"
    chip.faces.name="chip"
    chip.faces.col=(1,0,0)
    top.faces.name="top"
    geo = Glue([base,chip,top])
    geo.edges.name="default"
    geo.edges.Min(Y).name="bot"
    DrawGeo(geo)
    return OCCGeometry(geo, dim=2)

geo = MakeGeometryOCC()

Define the geometry by curves (old-style segments geometry):

#   point numbers 0, 1, ... 11
#   sub-domain numbers (1), (2), (3)
#
#
#             7-------------6
#             |             |
#             |     (2)     |
#             |             |
#      3------4-------------5------2
#      |                           |
#      |             11            |
#      |           /   \           |
#      |         10 (3) 9          |
#      |           \   /     (1)   |
#      |             8             |
#      |                           |
#      0---------------------------1
#

def MakeGeometry():
    from netgen.geom2d import SplineGeometry
    geometry = SplineGeometry()

    # point coordinates ...
    pnts = [ (0,0), (1,0), (1,0.6), (0,0.6), \
             (0.2,0.6), (0.8,0.6), (0.8,0.8), (0.2,0.8), \
             (0.5,0.15), (0.65,0.3), (0.5,0.45), (0.35,0.3) ]
    pnums = [geometry.AppendPoint(*p) for p in pnts]

    # start-point, end-point, boundary-condition, left-domain, right-domain:
    lines = [ (0,1,"bot",1,0), (1,2,"outer",1,0), (2,5,"outer",1,0), (5,4,"inner",1,2), (4,3,"outer",1,0), (3,0,"outer",1,0), \
              (5,6,"outer",2,0), (6,7,"outer",2,0), (7,4,"outer",2,0), \
              (8,9,"inner",3,1), (9,10,"inner",3,1), (10,11,"inner",3,1), (11,8,"inner",3,1) ]

    for p1,p2,bc,left,right in lines:
        geometry.Append(["line", pnums[p1], pnums[p2]], bc=bc, leftdomain=left, rightdomain=right)

    geometry.SetMaterial(1,"base")
    geometry.SetMaterial(2,"top")
    geometry.SetMaterial(3,"chip")

    return geometry

# geo = MakeGeometry()
# Draw(geo)

Piece-wise constant coefficients in sub-domains:

mesh = Mesh(geo.GenerateMesh(maxh=0.2))

fes = H1(mesh, order=3, dirichlet="bot", autoupdate=True)
u, v = fes.TnT()

lam = CoefficientFunction([1, 1000, 10])
a = BilinearForm(fes)
a += lam*grad(u)*grad(v)*dx

# heat-source in inner subdomain
f = LinearForm(fes)
f += 1*v*dx(definedon="chip")

c = Preconditioner(a, type="multigrid", inverse="sparsecholesky")

gfu = GridFunction(fes, autoupdate=True)

Assemble and solve problem:

def SolveBVP():
    a.Assemble()
    f.Assemble()
    inv = CGSolver(a.mat, c.mat)
    gfu.vec.data = inv * f.vec

SolveBVP()
Draw (gfu, mesh);

Gradient recovery error estimator: Interpolate finite element flux

\[q_h := I_h (\lambda \nabla u_h)\]

and take difference as element error indicator:

\[\eta_T := \tfrac{1}{\lambda} \| q_h - \lambda \nabla u_h \|_{L_2(T)}^2\]

l = []    # l = list of estimated total error
space_flux = HDiv(mesh, order=2, autoupdate=True)
gf_flux = GridFunction(space_flux, "flux", autoupdate=True)

def CalcError():

    # FEM-flux
    flux = lam * grad(gfu)
    # interpolate into H(div)
    gf_flux.Set(flux)

    # compute estimator:
    err = 1/lam*(flux-gf_flux)*(flux-gf_flux)
    eta2 = Integrate(err, mesh, VOL, element_wise=True)
    l.append ((fes.ndof, sqrt(sum(eta2))))
    print("ndof =", fes.ndof, " toterr =", sqrt(sum(eta2)))

    # mark for refinement:
    maxerr = max(eta2)
    # marking with Python loop:
    # for el in mesh.Elements():
    #    mesh.SetRefinementFlag(el, eta2[el.nr] > 0.25*maxerr)

    # marking using numpy:
    mesh.ngmesh.Elements2D().NumPy()["refine"] = \
        eta2.NumPy() > 0.25*maxerr

CalcError()

ndof = 208  toterr = 0.0062027218384066805

Adaptive loop:

level = 0
while fes.ndof < 50000:
    mesh.Refine()
    SolveBVP()
    CalcError()
    level = level+1
    if level%5 == 0:
        Draw (gfu)

ndof = 328  toterr = 0.005424419894719641
ndof = 583  toterr = 0.0034370983010913294
ndof = 997  toterr = 0.0024617055250623224
ndof = 1492  toterr = 0.0017417208970402825
ndof = 2143  toterr = 0.001159036769265472

ndof = 2917  toterr = 0.0007578273431000435
ndof = 3775  toterr = 0.000493096916783832
ndof = 4651  toterr = 0.0003247245805364616
ndof = 5440  toterr = 0.00021839607818093112
ndof = 6211  toterr = 0.00015253103584489278

ndof = 6910  toterr = 0.00011329794969631048
ndof = 7456  toterr = 9.041817442322814e-05
ndof = 8263  toterr = 7.05016031884768e-05
ndof = 9433  toterr = 5.622462330206224e-05
ndof = 10495  toterr = 4.587268141504037e-05

ndof = 12130  toterr = 3.351750230659081e-05
ndof = 13675  toterr = 2.564899368696983e-05
ndof = 15259  toterr = 2.077431963572974e-05
ndof = 18055  toterr = 1.5345303474890484e-05
ndof = 20347  toterr = 1.2390591822142242e-05

ndof = 23761  toterr = 9.324033982234093e-06
ndof = 27262  toterr = 7.144658685394859e-06
ndof = 31426  toterr = 5.654098735383225e-06
ndof = 37588  toterr = 4.121876481044875e-06
ndof = 43645  toterr = 3.159814850590886e-06

ndof = 50794  toterr = 2.4145846823594603e-06

Draw (gfu);

%matplotlib inline
import matplotlib.pyplot as plt
plt.yscale('log')
plt.xscale('log')
plt.xlabel("ndof")
plt.ylabel("H1 error-estimate")
ndof,err = zip(*l)
plt.plot(ndof,err, "-*")

plt.ion()
plt.show();