Tutorial 2: Dirichlet boundary conditions

Tutorial 2: Dirichlet boundary conditions#

Note: This tutorial is a minor adaption of the original Ngsolve i-tutorial on Dirichlet boundary conditions

This tutorial goes in depth into the mechanisms required to solve the Dirichlet problem

\[\begin{equation*} -\Delta u = f \quad \text{ in } \Omega, \end{equation*}\]

with a nonzero Dirichlet boundary condition

\[\begin{equation*} u|_{\Gamma_D} = g \quad \text{ on a boundary part } \Gamma_D \subset \partial\Omega. \end{equation*}\]

The same mechanisms are used in solving boundary value problems involving operators other than the Laplacian. You will see how to perform these tasks in NGSolve:

  • extend Dirichlet data from boundary parts,

  • convert boundary data into a volume source,

  • reduce inhomogeneous Dirichlet case to the homogeneous case, and

  • perform all these tasks automatically within a utility.

If you are interested only in an automatic utility that does all these steps under the hood, then please skip to the last part of the tutorial.

Spaces with Dirichlet conditions on part of the boundary#

from ngsolve import *
from ngsolve.webgui import Draw
# from netgen.geom2d import unit_square

mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))
mesh.GetBoundaries()

('bottom', 'right', 'top', 'left')

The unit_square has its boundaries marked as left, right, top and bottom. Suppose we want non-homogeneous Dirichlet boundary conditions on

\[ \Gamma_D = \Gamma_{left} \cup \Gamma_{right}. \]

Then, we set the space as follows:

fes = H1(mesh, order=2, dirichlet="left|right")

Compare this space with the one without the dirichlet flag:

fs2 = H1(mesh, order=2)
fes.ndof, fs2.ndof    # total number of unknowns

(133, 133)

Thus, the dirichlet flag did not change ndof. In NGSolve the unknowns are split into two groups:

  • dirichlet dofs (or constrained dofs),

  • free dofs.

The facility FreeDofs gives a BitArray such that FreeDofs[dof] is True if and only if dof is a free degree of freedom.

print("free dofs of fs2 without \"dirichlet\" flag:\n",
      fs2.FreeDofs())
print("free dofs of fes:\n", fes.FreeDofs())

free dofs of fs2 without "dirichlet" flag:
 0: 11111111111111111111111111111111111111111111111111
50: 11111111111111111111111111111111111111111111111111
100: 111111111111111111111111111111111
free dofs of fes:
 0: 00001111000011110000111111111111111111110101011011
50: 11111111110110110111111111111111011011011111111111
100: 111111111111111111111111111111111

  • The space fs2 without dirichlet flag has only free dofs (no dirichlet dofs).

  • The other space fes has a few dofs that are marked as not free. These are the dofs that are located on the boundary regions we marked as dirichlet.

Extension of boundary data#

We use the standard technique of reducing a problem with essential non-homogeneous boundary conditions to one with homogeneous boundary condition using an extension. The solution \(u\) in \(H^1\) satisfies

\[ u|_{\Gamma_D} = g\]

and

\[ \int_\Omega \nabla u \cdot \nabla v_0 = \int_\Omega f v_0 \]

for all \(v_0\) in \(\in H_{0,D}^1 = \{ v \in H^1: v|_{\Gamma_D} = 0\}\). Split the solution

\[ u = u_0 + u_D \]

where \(u_D\) is an extension of \(g\) into \(\Omega\). Then we only need to find \(u_0\) in \(H^1_{0,D}\) satisfying the homogeneous Dirichlet problem

\[ \int_\Omega \nabla u_0 \cdot \nabla v_0 = \int_\Omega f v_0 - \int_\Omega \nabla u_D \cdot \nabla v_0 \]

for all \(v_0\) in \(H_{0,D}^1\). These are the issues to consider in this approach:

  • How to define an extension \(u_D\) in the finite element space?

  • How to form and solve the system for \(u_0\)?

Let us address the first in the following example.

Suppose we are given that

\[\begin{equation*} g = \sin(y) \qquad \text{ on } \; \Gamma_D. \end{equation*}\]
g = sin(y)

We interpolate \(g\) on the boundary of the domain and extend it to zero on the elements not having an intersection with \(\Gamma_D\).

gfu = GridFunction(fes)
gfu.Set(g, BND)
# gfu.Set(g, definedon=mesh.Boundaries("left|right"))
Draw(gfu);

The keyword BND tells Set that g need only be interpolated on those parts of the boundary that are marked dirichlet.

Thus, gfu now contains the extension \(u_D\). Next, we turn to set up the system for \(u_0\).

Forms and assembly#

In NGSolve, bilinear and linear forms are defined independently of the dirichlet flags. Matrices and vectors are set up with respect to all unknowns (free or constrained) so they may be restricted to any group of unknowns later.

u, v = fes.TnT()
a = BilinearForm(grad(u)*grad(v)*dx).Assemble();

If \(A=\) a.mat is the matrix just assembled, then we want to solve for

\[ A (u_0 + u_D) = f \quad \Rightarrow \quad A u_0 = f - A u_D \]

or

\[\begin{split} \left( \begin{array}{cc} A_{FF} & A_{FD} \\ A_{DF} & A_{DD} \end{array} \right) \left( \begin{array}{c} u_{0,F} \\ 0 \end{array} \right) = \left( \begin{array}{c} {f}_F \\ {f}_D \end{array} \right) - \left( \begin{array}{cc} A_{FF} & A_{FD} \\ A_{DF} & A_{DD} \end{array} \right) \left( \begin{array}{c} u_{D,F} \\ u_{D,D} \end{array} \right) \end{split}\]

where we have block partitioned using free dofs (\(F\)) and dirichlet dofs (\(D\)) as if they were numbered consecutively (which may not be the case in practice) for ease of presentation. The first row gives

\[ A_{FF} u_{0,F} = f_F - [A u_D]_F. \]

Since we have already constructed \(u_D\), we need to perform these next steps:

  • Set up the right hand side from \(f\) and \(u_D\).

  • Solve a linear system which involves only \(A_{FF}\).

  • Add solution: \(u = u_0 + u_D\).

Solve for the free dofs#

We need to assemble the right hand side of \(A_{FF} u_{0,F} = f_F - [A u_D]_F\), namely

\[ r = f - A u_D. \]

f = LinearForm(1*v*dx).Assemble()
r = f.vec - a.mat * gfu.vec

The implementation of

\[\begin{split} u = u_D + \left( \begin{array}{cc} A_{FF}^{-1} & 0 \\ 0 & 0 \end{array} \right) r \end{split}\]

by sparse solvers is achieved by the following:

gfu.vec.data += a.mat.Inverse(freedofs=fes.FreeDofs()) * r  
Draw(gfu);

The automatic utility BVP#

NGSolve also provides a BVP facility in the solvers submodule, within which the above steps are performed automatically. You provide \(A\), \(f\), a grid function gfu with your boundary condition \(g\), and optionally a preconditioner. Then BVP solves the problem with non-homogeneous Dirichlet boundary condition and overwrites gfu with the solution.

gfu.Set(g, BND)
solvers.BVP(bf=a, lf=f, gf=gfu)
Draw(gfu)

BaseWebGuiScene
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